Search: a003338 -id:a003338
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A003072
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Numbers that are the sum of 3 positive cubes.
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+10
85
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3, 10, 17, 24, 29, 36, 43, 55, 62, 66, 73, 80, 81, 92, 99, 118, 127, 129, 134, 136, 141, 153, 155, 160, 179, 190, 192, 197, 216, 218, 225, 232, 244, 251, 253, 258, 270, 277, 281, 288, 307, 314, 342, 344, 345, 349, 352, 359, 368, 371, 375, 378, 397, 405, 408, 415, 433, 434
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OFFSET
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1,1
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COMMENTS
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LINKS
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FORMULA
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EXAMPLE
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a(11) = 73 = 1^3 + 2^3 + 4^3, which is sum of three cubes.
a(15) = 99 = 2^3 + 3^3 + 4^3, which is sum of three cubes.
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MAPLE
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isA003072 := proc(n)
local x, y, z;
for x from 1 do
if 3*x^3 > n then
return false;
end if;
for y from x do
if x^3+2*y^3 > n then
break;
end if;
if isA000578(n-x^3-y^3) then
return true;
end if;
end do:
end do:
end proc:
for n from 1 to 1000 do
if isA003072(n) then
printf("%d, ", n) ;
end if;
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MATHEMATICA
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Select[Range[435], (p = PowersRepresentations[#, 3, 3]; (Select[p, #[[1]] > 0 && #[[2]] > 0 && #[[3]] > 0 &] != {})) &] (* Jean-François Alcover, Apr 29 2011 *)
With[{upto=500}, Select[Union[Total/@Tuples[Range[Floor[Surd[upto-2, 3]]]^3, 3]], #<=upto&]] (* Harvey P. Dale, Oct 25 2021 *)
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PROG
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(PARI) sum(n=1, 11, x^(n^3), O(x^1400))^3 /* Then [i|i<-[1..#%], polcoef(%, i)] gives the list of powers with nonzero coefficient. - M. F. Hasler, Aug 02 2020 */
(PARI) list(lim)=my(v=List(), k, t); lim\=1; for(x=1, sqrtnint(lim-2, 3), for(y=1, min(sqrtnint(lim-x^3-1, 3), x), k=x^3+y^3; for(z=1, min(sqrtnint(lim-k, 3), y), listput(v, k+z^3)))); Set(v) \\ Charles R Greathouse IV, Sep 14 2015
(Haskell)
a003072 n = a003072_list !! (n-1)
a003072_list = filter c3 [1..] where
c3 x = any (== 1) $ map (a010057 . fromInteger) $
takeWhile (> 0) $ map (x -) $ a003325_list
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CROSSREFS
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Cf. A###### (x, y) = Numbers that are the sum of x nonzero y-th powers: A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A003336
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Numbers that are the sum of 2 positive 4th powers.
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+10
77
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2, 17, 32, 82, 97, 162, 257, 272, 337, 512, 626, 641, 706, 881, 1250, 1297, 1312, 1377, 1552, 1921, 2402, 2417, 2482, 2592, 2657, 3026, 3697, 4097, 4112, 4177, 4352, 4721, 4802, 5392, 6497, 6562, 6577, 6642, 6817, 7186, 7857, 8192, 8962, 10001, 10016, 10081, 10256, 10625
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OFFSET
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1,1
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COMMENTS
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Numbers k such that k = x^4 + y^4 has a solution in positive integers x, y.
There are no squares in this sequence. - Altug Alkan, Apr 08 2016
As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020
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LINKS
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FORMULA
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EXAMPLE
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16378801 is in the sequence as 16378801 = 43^4 + 60^4.
39126977 is in the sequence as 39126977 = 49^4 + 76^4.
71769617 is in the sequence as 71769617 = 19^4 + 92^4. (End)
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MATHEMATICA
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nn=12; Select[Union[Plus@@@(Tuples[Range[nn], {2}]^4)], # <= nn^4&] (* Harvey P. Dale, Dec 29 2010 *)
Select[Range@ 11000, Length[PowersRepresentations[#, 2, 4] /. {0, _} -> Nothing] > 0 &] (* Michael De Vlieger, Apr 08 2016 *)
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PROG
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(PARI) list(lim)=my(v=List()); for(x=1, sqrtnint(lim\=1, 4), for(y=1, min(sqrtnint(lim-x^4, 4), x), listput(v, x^4+y^4))); Set(v) \\ Charles R Greathouse IV, Apr 24 2012; updated July 13 2024
(PARI) T=thueinit('x^4+1, 1);
(Python)
def aupto(lim):
p1 = set(i**4 for i in range(1, int(lim**.25)+2) if i**4 <= lim)
p2 = set(a+b for a in p1 for b in p1 if a+b <= lim)
return sorted(p2)
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CROSSREFS
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5906 is the first term in A060387 but not in this sequence. Cf. A020897.
Cf. A088687 (2 distinct 4th powers).
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A000337
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a(n) = (n-1)*2^n + 1.
(Formerly M3874 N1587)
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+10
69
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0, 1, 5, 17, 49, 129, 321, 769, 1793, 4097, 9217, 20481, 45057, 98305, 212993, 458753, 983041, 2097153, 4456449, 9437185, 19922945, 41943041, 88080385, 184549377, 385875969, 805306369, 1677721601, 3489660929, 7247757313, 15032385537, 31138512897, 64424509441
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OFFSET
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0,3
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COMMENTS
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a(n) also gives number of 0's in binary numbers 1 to 111..1 (n+1 bits). - Stephen G Penrice, Oct 01 2000
a(n) is the number of directed column-convex polyominoes of area n+2 having along the lower contour exactly one vertical step that is followed by a horizontal step (a reentrant corner). - Emeric Deutsch, May 21 2003
a(n) is the number of bits in binary numbers from 1 to 111...1 (n bits). Partial sums of A001787. - Emeric Deutsch, May 24 2003
Genus of graph of n-cube = a(n-3) = 1+(n-4)*2^(n-3), n>1.
Sum of ordered partitions of n where each element is summed via T(e-1). See A066185 for more information. - Jon Perry, Dec 12 2003
a(n-2) is the number of Dyck n-paths with exactly one peak at height >= 3. For example, there are 5 such paths with n=4: UUUUDDDD, UUDUUDDD, UUUDDUDD, UDUUUDDD, UUUDDDUD. - David Callan, Mar 23 2004
Permutations in S_{n+2} avoiding 12-3 that contain the pattern 13-2 exactly once.
a(n) is prime for n = 2, 3, 7, 27, 51, 55, 81. a(n) is semiprime for n = 4, 5, 6, 8, 9, 10, 11, 13, 15, 19, 28, 32, 39, 57, 63, 66, 75, 97. - Jonathan Vos Post, Jul 18 2005
A member of the family of sequences defined by a(n) = Sum_{i=1..n} i*[c(1)*...*c(r)]^(i-1). This sequence has c(1)=2, A014915 has c(1)=3. - Ctibor O. Zizka, Feb 23 2008
Starting with 1 = row sums of A023758 as a triangle by rows: [1; 2,3; 4,6,7; 8,12,14,15; ...]. - Gary W. Adamson, Jul 18 2008
Equivalent formula given in Brehm: for each q >= 3 there exists a polyhedral map M_q of type {4, q} with [number of vertices] f_0 = 2^q and [genus] g = (2^(q-3))*(q-4) + 1 such that M_q and its dual have polyhedral embeddings in R^3 [McMullen et al.]. - Jonathan Vos Post, Jul 25 2009
(1 + 5*x + 17*x^2 + 49*x^3 + ...) = (1 + 2*x + 4*x^2 + 8*x^3 + ...) * (1 + 3*x + 7*x^2 + 15*x^3 + ...). - Gary W. Adamson, Mar 14 2012
The first barycentric coordinate of the centroid of Pascal triangles, assuming that numbers are weights, is A000295(n+1)/A000337(n), no matter what the triangle sides are. See attached figure. - César Eliud Lozada, Nov 14 2014
a(n) is the n-th number that is a sum of n positive n-th powers for n >= 1. a(4) = 49 = A003338(4). - Alois P. Heinz, Aug 01 2020
a(n) is the sum of the largest elements of all subsets of {1,2,..,n}. For example, a(3)=17; the subsets of {1,2,3} are {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}, and the sum of the largest elements is 17. - Enrique Navarrete, Aug 20 2020
a(n-1) is the sum of the second largest elements of the subsets of {1,2,..,n} that contain n. For example, for n = 4, a(3)=17; the subsets of {1,2,3,4} that contain 4 are {4}, {1,4}, {2,4}, {3,4}, {1,2,4}, {1,3,4}, {2,3,4}, {1,2,3,4}, and the sum of the second largest elements is 17. - Enrique Navarrete, Aug 24 2020
a(n-1) is also the sum of diameters of all subsets of {1,2,...,n} that contain n. For example, for n = 4, a(3)=17; the subsets of {1,2,3,4} that contain 4 are {4}, {1,4}, {2,4}, {3,4}, {1,2,4}, {1,3,4}, {2,3,4}, {1,2,3,4}; the diameters of these sets are 0,3,2,1,3,3,2,3 and the sum is 17. - Enrique Navarrete, Sep 07 2020
a(n-1) is also the number of additions required to compute the permanent of general n X n matrices using trellis methods (see Theorems 5 and 6, pp. 10-11 in Kiah et al.). - Stefano Spezia, Nov 02 2021
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REFERENCES
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F. Harary, Topological concepts in graph theory, pp. 13-17 of F. Harary and L. Beineke, editors, A seminar on Graph Theory, Holt, Rinehart and Winston, New York, 1967.
V. G. Gutierrez and S. L. de Medrano, Surfaces as complete intersections, in Riemann and Klein Surfaces, Automorphisms, Symmetries and Moduli Spaces, edited by Milagros Izquierdo, S. Allen Broughton, Antonio F. Costa, Contemp. Math. vol. 629, 2014, pp. 171-.
F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 119.
G. H. Hardy, A Theorem Concerning the Infinite Cardinal Numbers, Quart. J. Math., 35 (1904), p. 90 = Collected Papers of G. H. Hardy, Vol. VII, p. 430.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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Binomial transform of A004273. Binomial transform of A008574 if the leading zero is dropped.
G.f.: x/((1-x)*(1-2*x)^2). - Simon Plouffe in his 1992 dissertation
E.g.f.: exp(x) - exp(2*x)*(1-2*x). a(n) = 4*a(n-1) - 4*a(n-2)+1, n>0. Series reversion of g.f. A(x) is x*A034015(-x). - Michael Somos
Binomial transform of n/(n+1) is a(n)/(n+1). - Paul Barry, Aug 19 2005
Convolution of "Number of fixed points in all 231-avoiding involutions in S_n" (A059570) with "The odd numbers" (A005408), treating the result as if offset=0. - Graeme McRae, Jul 12 2006
a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3), n > 2. - Harvey P. Dale, Jun 21 2011
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MAPLE
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MATHEMATICA
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Table[Sum[(-1)^(n - k) k (-1)^(n - k) Binomial[n + 1, k + 1], {k, 0, n}], {n, 0, 28}] (* Zerinvary Lajos, Jul 08 2009 *)
LinearRecurrence[{5, -8, 4}, {0, 1, 5}, 40] (* Harvey P. Dale, Jun 21 2011 *)
CoefficientList[Series[x / ((1 - x) (1 - 2 x)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Nov 21 2014 *)
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PROG
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(PARI) a(n)=if(n<0, 0, (n-1)*2^n+1)
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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A003327
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Numbers that are the sum of 4 positive cubes in 1 or more way.
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+10
57
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4, 11, 18, 25, 30, 32, 37, 44, 51, 56, 63, 67, 70, 74, 81, 82, 88, 89, 93, 100, 107, 108, 119, 126, 128, 130, 135, 137, 142, 144, 145, 149, 154, 156, 161, 163, 168, 180, 182, 187, 191, 193, 198, 200, 205, 206, 217, 219, 224, 226, 233, 240, 243, 245, 252, 254
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OFFSET
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1,1
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COMMENTS
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It is conjectured that every number greater than 7373170279850 is in this sequence. [See the paper of the same name. - T. D. Noe, May 25 2017] - Charles R Greathouse IV, Jan 14 2017
As the order of addition doesn't matter we can assume terms are in increasing order. - David A. Corneth, Aug 01 2020
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LINKS
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Jean-Marc Deshouillers, François Hennecart, Bernard Landreau, 7373170279850, Math. Comp. 69 (2000), pp. 421-439. Appendix by I. Gusti Putu Purnaba.
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EXAMPLE
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3888 is in the sequence as 3888 = 6^3 + 6^3 + 12^3 + 12^3.
7729 is in the sequence as 7729 = 2^3 + 4^3 + 14^3 + 17^3.
7875 is in the sequence as 7875 = 5^3 + 10^3 + 15^3 + 15^3. (End)
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PROG
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(PARI) list(lim)=my(v=List(), e=1+lim\1, x='x, t); t=sum(i=1, sqrtnint(e-4, 3), x^i^3, O(x^e))^4; for(n=4, lim, if(polcoeff(t, n)>0, listput(v, n))); Vec(v) \\ Charles R Greathouse IV, Jan 14 2017
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CROSSREFS
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A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A000414
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Numbers that are the sum of 4 nonzero squares.
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+10
51
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4, 7, 10, 12, 13, 15, 16, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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As the order of addition doesn't matter we can assume terms are in increasing order. - David A. Corneth, Aug 01 2020
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LINKS
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FORMULA
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EXAMPLE
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1608 is in the sequence as 1608 = 18^2 + 20^2 + 20^2 + 22^2.
2140 is in the sequence as 2140 = 21^2 + 21^2 + 23^2 + 27^2.
3298 is in the sequence as 3298 = 25^2 + 26^2 + 29^2 + 34^2. (End)
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MATHEMATICA
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q=16; lst={}; Do[Do[Do[Do[z=a^2+b^2+c^2+d^2; If[z<=(q^2)+3, AppendTo[lst, z]], {d, q}], {c, q}], {b, q}], {a, q}]; Union@lst (*Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)
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PROG
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(PARI) is(n)=my(k=if(n, n/4^valuation(n, 4), 2)); k!=2 && k!=6 && k!=14 && !setsearch([0, 1, 3, 5, 9, 11, 17, 29, 41], n) \\ Charles R Greathouse IV, Sep 03 2014
(Python)
limit = 10026 # 10000th term in b-file
from functools import lru_cache
nzs = [k*k for k in range(1, int(limit**.5)+2) if k*k + 3 <= limit]
nzss = set(nzs)
@lru_cache(maxsize=None)
def ok(n, m): return n in nzss if m == 1 else any(ok(n-s, m-1) for s in nzs)
(Python)
from itertools import count, islice
def A000414_gen(startvalue=0): # generator of terms >= startvalue
return filter(lambda n:not(n in {0, 1, 3, 5, 9, 11, 17, 29, 41} or n>>((~n&n-1).bit_length()&-2) in {2, 6, 14}), count(max(startvalue, 0)))
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CROSSREFS
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A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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corrected 6/95
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STATUS
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approved
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A003328
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Numbers that are the sum of 5 positive cubes.
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+10
49
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5, 12, 19, 26, 31, 33, 38, 40, 45, 52, 57, 59, 64, 68, 71, 75, 78, 82, 83, 89, 90, 94, 96, 97, 101, 108, 109, 115, 116, 120, 127, 129, 131, 134, 135, 136, 138, 143, 145, 146, 150, 152, 153, 155, 157, 162, 164, 169, 171, 172, 176, 181, 183, 188, 190, 192, 194, 195, 199
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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As the order of addition doesn't matter we can assume terms are in increasing order. - David A. Corneth, Aug 01 2020
It seems only a finite number N of positive integers are not in this sequence, and thus a(n) = n - N for all sufficiently large n. Is it true that 2243453, last term of A048927, is sufficiently large in that sense? - M. F. Hasler, Jan 04 2023
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LINKS
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EXAMPLE
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3084 is in the sequence as 3084 = 5^3 + 5^3 + 5^3 + 8^3 + 13^3.
4385 is in the sequence as 4385 = 4^3 + 4^3 + 9^3 + 11^3 + 13^3.
5426 is in the sequence as 5426 = 8^3 + 9^3 + 9^3 + 12^3 + 12^3. (End)
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PROG
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(PARI) select( {is_A003328(n, k=5, m=3, L=sqrtnint(abs(n-k+1), m))=if( n>k*L^m || n<k, 0, n<k*L^m, forstep(r=min(k-1, n\L^m), 0, -1, self()(n-r*L^m, k-r, m, L-1) && return(1)), 1)}, [1..200]) \\ M. F. Hasler, Aug 02 2020
A003328_upto(N, k=5, m=3)=[i|i<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, i)] \\ M. F. Hasler, Aug 02 2020
(Python)
from collections import Counter
from itertools import combinations_with_replacement as combs_w_rep
def aupto(lim):
s = filter(lambda x: x<=lim, (i**3 for i in range(1, int(lim**(1/3))+2)))
s2 = filter(lambda x: x<=lim, (sum(c) for c in combs_w_rep(s, 5)))
s2counts = Counter(s2)
return sorted(k for k in s2counts)
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CROSSREFS
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Cf. A###### (x, y) = Numbers that are the sum of x nonzero y-th powers:
A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A003349
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Numbers that are the sum of 4 positive 5th powers.
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+10
49
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4, 35, 66, 97, 128, 246, 277, 308, 339, 488, 519, 550, 730, 761, 972, 1027, 1058, 1089, 1120, 1269, 1300, 1331, 1511, 1542, 1753, 2050, 2081, 2112, 2292, 2323, 2534, 3073, 3104, 3128, 3159, 3190, 3221, 3315, 3370, 3401, 3432, 3612, 3643, 3854, 4096, 4151, 4182, 4213
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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LINKS
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EXAMPLE
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670593 is in the sequence as 670593 = 1^5 + 8^5 + 10^5 + 14^5.
862512 is in the sequence as 862512 = 7^5 + 9^5 + 12^5 + 14^5.
1892695 is in the sequence as 1892695 = 1^5 + 1^5 + 5^5 + 18^5. (End)
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MATHEMATICA
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f@n_:= Select[Range@n, IntegerPartitions[#, {4}, Range@(n^(1/5))^5] != {} &]; f@10000 (* Hans Rudolf Widmer, Dec 04 2022 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A003337
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Numbers n which are the sum of 3 nonzero 4th powers.
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+10
46
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3, 18, 33, 48, 83, 98, 113, 163, 178, 243, 258, 273, 288, 338, 353, 418, 513, 528, 593, 627, 642, 657, 707, 722, 768, 787, 882, 897, 962, 1137, 1251, 1266, 1298, 1313, 1328, 1331, 1378, 1393, 1458, 1506, 1553, 1568, 1633, 1808, 1875, 1922, 1937, 2002, 2177
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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Numbers which are in this sequence but not in A047714 must also be the sum of 2 biquadrates, or equal to a fourth power. Among the first 1000 terms of this sequence, this is the case for 4802 = 2*7^4, 57122 = 2*13^4 and 76832 = 2*14^4. - M. F. Hasler, Dec 31 2012
As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020
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LINKS
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EXAMPLE
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194818 is in the sequence as 194818 = 3^4 + 4^4 + 21^4.
480113 is in the sequence as 480113 = 7^4 + 12^4 + 26^4.
693842 is in the sequence as 693842 = 13^4 + 15^4 + 28^4. (End)
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PROG
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(Python)
def aupto(lim):
p1 = set(i**4 for i in range(1, int(lim**.25)+2) if i**4 <= lim)
p2 = set(a+b for a in p1 for b in p1 if a+b <= lim)
p3 = set(apb+c for apb in p2 for c in p1 if apb+c <= lim)
return sorted(p3)
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CROSSREFS
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A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A003358
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Numbers that are the sum of 2 nonzero 6th powers.
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+10
44
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2, 65, 128, 730, 793, 1458, 4097, 4160, 4825, 8192, 15626, 15689, 16354, 19721, 31250, 46657, 46720, 47385, 50752, 62281, 93312, 117650, 117713, 118378, 121745, 133274, 164305, 235298, 262145, 262208, 262873, 266240, 277769, 308800, 379793, 524288
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020
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LINKS
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EXAMPLE
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10069120217 is in the sequence as 10069120217 = 29^6 + 46^6.
139314070233 is in the sequence as 139314070233 = 3^6 + 72^6.
404680615040 is in the sequence as 404680615040 = 22^6 + 86^6. (End)
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MATHEMATICA
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CROSSREFS
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A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A003380
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Numbers that are the sum of 2 nonzero 8th powers.
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+10
44
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2, 257, 512, 6562, 6817, 13122, 65537, 65792, 72097, 131072, 390626, 390881, 397186, 456161, 781250, 1679617, 1679872, 1686177, 1745152, 2070241, 3359232, 5764802, 5765057, 5771362, 5830337, 6155426, 7444417, 11529602, 16777217, 16777472, 16783777, 16842752
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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LINKS
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EXAMPLE
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274893519322337 is in the sequence as 274893519322337 = 58^8 + 59^8.
357707312890625 is in the sequence as 357707312890625 = 50^8 + 65^8.
2590188068194497 is in the sequence as 2590188068194497 = 57^8 + 84^8. (End)
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MAPLE
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local a, x, x8, y, y8 ;
a := {} ;
for x from 1 do
x8 := x^8 ;
if 2*x8 > nmax then
break;
end if;
for y from x do
y8 := y^8 ;
if x8+y8 > nmax then
break;
end if;
if x8+y8 <= nmax then
a := a union {x8+y8} ;
end if;
end do:
end do:
sort(convert(a, list)) ;
end proc:
nmax := 20000000000000000 ;
LISTTOBFILE(L, "b003380.txt", 1) ; # R. J. Mathar, Aug 01 2020
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MATHEMATICA
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PROG
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(PARI) list(lim)=my(v=List(), x8); for(x=1, sqrtnint(lim\=1, 8), x8=x^8; for(y=1, min(sqrtnint(lim-x8, 8), x), listput(v, x8+y^8))); Set(v) \\ Charles R Greathouse IV, Aug 22 2017
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CROSSREFS
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A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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