Displaying 81-90 of 393 results found.
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Sums of 6 distinct powers of 10.
+10
10
111111, 1011111, 1101111, 1110111, 1111011, 1111101, 1111110, 10011111, 10101111, 10110111, 10111011, 10111101, 10111110, 11001111, 11010111, 11011011, 11011101, 11011110, 11100111, 11101011, 11101101, 11101110, 11110011, 11110101, 11110110, 11111001, 11111010
MATHEMATICA
Sort[Plus @@@ Subsets[10^Range[0, 7], {6}]] (* Amiram Eldar, Jul 12 2022 *)
Sums of 7 distinct powers of 10.
+10
10
1111111, 10111111, 11011111, 11101111, 11110111, 11111011, 11111101, 11111110, 100111111, 101011111, 101101111, 101110111, 101111011, 101111101, 101111110, 110011111, 110101111, 110110111, 110111011, 110111101, 110111110, 111001111, 111010111, 111011011, 111011101
MATHEMATICA
Take[Total/@Subsets[10^Range[0, 20], {7}]//Union, 20] (* Harvey P. Dale, Feb 25 2018 *)
Sums of 11 distinct powers of 10.
+10
10
11111111111, 101111111111, 110111111111, 111011111111, 111101111111, 111110111111, 111111011111, 111111101111, 111111110111, 111111111011, 111111111101, 111111111110, 1001111111111, 1010111111111, 1011011111111, 1011101111111, 1011110111111, 1011111011111, 1011111101111
MATHEMATICA
Union[Total/@Subsets[10^Range[0, 12], {11}]] (* Harvey P. Dale, Jan 20 2013 *)
PROG
(PARI) lista(nn) = {for (n=1, nn, if (hammingweight(n) == 11, print1(subst(Pol(binary(n)), x, 10), ", "); ); ); } \\ Michel Marcus, Feb 29 2016
Sums of 12 distinct powers of 10.
+10
10
111111111111, 1011111111111, 1101111111111, 1110111111111, 1111011111111, 1111101111111, 1111110111111, 1111111011111, 1111111101111, 1111111110111, 1111111111011, 1111111111101, 1111111111110, 10011111111111, 10101111111111, 10110111111111, 10111011111111, 10111101111111
MAPLE
N:= 14: # to get all terms of at most N digits
sort(map(t -> (10^N-1)/9 - add(10^j, j=t),
combinat:-choose([$0..N-1], N-12))); # Robert Israel, Feb 28 2016
MATHEMATICA
Sort[Plus @@@ Subsets[10^Range[0, 12], {12}]] (* Amiram Eldar, Jul 12 2022 *)
3, 21, 201, 2001, 20001, 200001, 2000001, 20000001, 200000001, 2000000001, 20000000001, 200000000001, 2000000000001, 20000000000001, 200000000000001, 2000000000000001, 20000000000000001, 200000000000000001, 2000000000000000001
COMMENTS
Numbers k such that (R(k) - 1)/(k + 1) = 1/2, where R(k) denotes the digit reversal of k (cf. A004086). - Stefano Spezia, Nov 25 2023
FORMULA
a(n) = 10*a(n-1)-9.
a(n) = 11*a(n-1)-10*a(n-2).
G.f.: 3*(1-4*x)/((1-x)*(1-10*x)).
MATHEMATICA
NestList[10#-9&, 3, 20] (* or *) LinearRecurrence[{11, -10}, {3, 21}, 20] (* Harvey P. Dale, Sep 30 2017 *)
PROG
(Magma) [2*10^n+1: n in [0..30]];
(Haskell)
Rectangular array read by upwards antidiagonals: a(n,k) = index of largest term <= 10^k in row n of A253572, n >= 1, k >= 0.
+10
10
1, 1, 4, 1, 7, 7, 1, 9, 20, 10, 1, 10, 34, 40, 14, 1, 10, 46, 86, 67, 17, 1, 10, 55, 141, 175, 101, 20, 1, 10, 62, 192, 338, 313, 142, 24, 1, 10, 67, 242, 522, 694, 507, 190, 27, 1, 10, 72, 287, 733, 1197, 1273, 768, 244, 30
COMMENTS
Or a(n,k) = the number of positive integers less than or equal to 10^k that are divisible by no prime exceeding prime(n).
EXAMPLE
Array begins:
{1, 4, 7, 10, 14, 17, 20, 24, 27, 30, ...}
{1, 7, 20, 40, 67, 101, 142, 190, 244, 306, ...}
{1, 9, 34, 86, 175, 313, 507, 768, 1105, 1530, ...}
{1, 10, 46, 141, 338, 694, 1273, 2155, 3427, 5194, ...}
{1, 10, 55, 192, 522, 1197, 2432, 4520, 7838, 12867, ...}
{1, 10, 62, 242, 733, 1848, 4106, 8289, 15519, 27365, ...}
{1, 10, 67, 287, 945, 2579, 6179, 13389, 26809, 50351, ...}
{1, 10, 72, 331, 1169, 3419, 8751, 20198, 42950, 85411, ...}
{1, 10, 76, 369, 1385, 4298, 11654, 28434, 63768, 133440, ...}
{1, 10, 79, 402, 1581, 5158, 14697, 37627, 88415, 193571, ...}
MATHEMATICA
r = 10; y[1] = t = Table[2^j, {j, 0, 39}]; max = 10^13; len = 10^10; prev = 0; For[n = 2, n <= r, n++, next = 0; For[k = 1, k <= 43, k++, If[Prime[n]^k < max, t = Union[t, Prime[n]*t]; s = FirstPosition[t, v_ /; v > len, 0]; t = Take[t, s[[1]] - 1]; If[t[[-1]] > len, t = Delete[t, -1]]; next = Length[t]; If[next == prev, Break, prev = next], Break]]; y[n] = t]; b[i_, j_] := FirstPosition[y[i], v_ /; v > 10^j][[1]]; a253635[n_, j_] := If[IntegerQ[b[n, j]], b[n, j] - 1, 0]; Flatten[Table[a253635[n - j, j], {n, r}, {j, 0, n - 1}]] (* array antidiagonals flattened *)
Numbers k such that the sum of digits of 9k is 27.
+10
10
111, 211, 221, 222, 311, 321, 322, 331, 332, 333, 411, 421, 422, 431, 432, 433, 441, 442, 443, 444, 511, 521, 522, 531, 532, 533, 541, 542, 543, 544, 551, 552, 553, 554, 555, 611, 621, 622, 631, 632, 633, 641, 642, 643, 644, 651, 652, 653, 654, 655, 661
COMMENTS
The digital sum of 9k is always a multiple of 9. For most numbers below 100 it is actually equal to 9. Numbers such that the digital sum of 9k is 18 are listed in A279769. Only every third term of the present sequence is divisible by 3. The sequence of record gaps [and upper end of the gap] is: 100 [a(2) = 211], 101 [a(221) = 1211], 111 [a(4841) = 11211], 111 [a(10121) = 22311], 111 [a(15752) = 33411], ..., 111 [a(45133) = 88911], 111 [a(50413) = 100011], 211 [a(55253) = 111211], 311 [a(110000) = 222311], ..., 911 [a(380557) = 888911], 1011 [a(411049) = 1000011], 1211 [a(436976) = 1111211], 2311 [a(840281) = 2222311], ..., 8911 [a(2451241) = 8888911], ...
MATHEMATICA
Select[Range@ 661, Total@ IntegerDigits[9 #] == 27 &] (* Michael De Vlieger, Dec 23 2016 *)
PROG
(PARI) is(n)=sumdigits(9*n)==27
CROSSREFS
Numbers with given digital sum: A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225 (14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
a(n) = (10^(2n+1)-1)/3 - 3*10^n.
+10
10
0, 303, 33033, 3330333, 333303333, 33333033333, 3333330333333, 333333303333333, 33333333033333333, 3333333330333333333, 333333333303333333333, 33333333333033333333333, 3333333333330333333333333, 333333333333303333333333333, 33333333333333033333333333333, 3333333333333330333333333333333
FORMULA
G.f.: 3*x*(101 - 200*x)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
E.g.f.: exp(x)*(10*exp(99*x) - 9*exp(9*x) - 1)/3. - Stefano Spezia, Jul 13 2024
MAPLE
A332130 := n -> (10^(2*n+1)-1)/3-3*10^n;
MATHEMATICA
Array[ (10^(2 # + 1)-1)/3 - 3*10^#) &, 15, 0]
PROG
(PARI) apply( { A332130(n)=10^(n*2+1)\3-3*10^n}, [0..15]) (Python) def A332130(n): return 10**(n*2+1)//3-3*10**n
CROSSREFS
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes). Cf. A332120 .. A332190 (variants with different repeated digit 2, ..., 9). Cf. A332131 .. A332139 (variants with different middle digit 1, ..., 9).
a(n+1) = 8*a(n) + 9*a(n-1), a(0) = 0, a(1) = 1.
+10
9
0, 1, 8, 73, 656, 5905, 53144, 478297, 4304672, 38742049, 348678440, 3138105961, 28242953648, 254186582833, 2287679245496, 20589113209465, 185302018885184, 1667718169966657, 15009463529699912, 135085171767299209, 1215766545905692880, 10941898913151235921
COMMENTS
Number of walks of length n between any two distinct nodes of the complete graph K_10. Example: a(2) = 8 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHIJ are: ACB, ADB, AEB, AFB, AGB, AHB, AIB and AJB. - Emeric Deutsch, Apr 01 2004 The ratio a(n+1)/a(n) converges to 9 as n approaches infinity. - Felix P. Muga II, Mar 09 2014
FORMULA
G.f.: x/((1+x)*(1-9*x)).
E.g.f. exp(4*x)*sinh(5*x)/5.
a(n) = (9^n - (-1)^n)/10. (End)
MATHEMATICA
Table[(9^n - (-1)^n)/10, {n, 0, 30}] (* or *) LinearRecurrence[{8, 9}, {0, 1}, 30] (* G. C. Greubel, Jan 06 2018 *)
PROG
(PARI) A015577_vec(N=20)=Vec(O(x^N)+1/(1-8*x-9*x^2), -N-1) \\ M. F. Hasler, Jun 14 2008, edited Oct 25 2019 (PARI) for(n=0, 30, print1((9^n - (-1)^n)/10, ", ")) \\ G. C. Greubel, Jan 06 2018 (Sage) [lucas_number1(n, 8, -9) for n in range(0, 19)] # Zerinvary Lajos, Apr 25 2009 (Maxima)
a[0]:0$
a[n]:=9^(n-1)-a[n-1]$
CROSSREFS
Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008
Expansion of 1/((1-2x)(1-10x)).
+10
9
1, 12, 124, 1248, 12496, 124992, 1249984, 12499968, 124999936, 1249999872, 12499999744, 124999999488, 1249999998976, 12499999997952, 124999999995904, 1249999999991808, 12499999999983616, 124999999999967232
FORMULA
a(n) = 125*10^(n-2) - 2^(n-2) = a(n-1)*10 + 2^n. - Henry Bottomley, Jun 06 2000
EXAMPLE
G.f. = 1 + 12*x + 124*x^2 + 1248*x^3 + 12496*x^4 + 124992*x^5 + ...
MATHEMATICA
f[n_] := Sum[2^(k - 1)*10^(n - k), {k, n}]; Array[f, 18] (* Robert G. Wilson v, Dec 03 2016 *) LinearRecurrence[{12, -20}, {1, 12}, 20] (* Harvey P. Dale, Jul 28 2020 *)
PROG
(Sage) [lucas_number1(n, 12, 20) for n in range(1, 18)] # Zerinvary Lajos, Apr 27 2009 (Sage) [(10^n - 2^n)/8 for n in range(1, 19)] # Zerinvary Lajos, Jun 05 2009
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