Displaying 1-10 of 17 results found.
Corresponding values of arithmetic means of divisors of numbers from A007340.
+20
1
1, 3, 28, 45, 84, 182, 270, 620, 546, 1240, 1330, 1638, 1260, 1365, 2660, 8128, 6200, 6200, 6944, 8190, 9310, 7560, 8190, 18600, 15120, 18620, 16380, 25935, 30240, 32760, 63700, 55860, 40950, 55800, 117800, 98208, 105664, 117800, 121030, 173600, 167400, 155610
COMMENTS
A007340 = numbers k such that the arithmetic mean of divisors of k is integer and equal to one of the divisors of this numbers.
EXAMPLE
a(3) = 28, because 28 is arithmetic mean of divisors of A007340(3)=140 and also is divisor of 140.
PROG
[SumOfDivisors(n) / NumberOfDivisors(n): n in[ A007340(n)] ] (PARI) lista(kmax) = {my(f, s, d); for(k = 1, kmax, f = factor(k); s = sigma(f); d = numdiv(f); if(!(s % d) && denominator(d / sigma(f, -1)) == 1, print1(s/d, ", "))); } \\ Amiram Eldar, Apr 15 2024
1, 6, 140, 270, 672, 1638, 2970, 6200, 8190, 18600, 18620, 27846, 30240, 32760, 55860, 105664
Numbers j such that the average of the divisors of j is an integer: sigma_0(j) divides sigma_1(j). Alternatively, numbers j such that tau(j) ( A000005(j)) divides sigma(j) ( A000203(j)).
(Formerly M2389)
+10
117
1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 77, 78, 79, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 99, 101, 102, 103, 105
COMMENTS
Sometimes called arithmetic numbers.
Generalized (sigma_r)-numbers are numbers j for which sigma_r(j)/sigma_0(j) = c^r. Sigma_r(j) denotes the sum of the r-th powers of the divisors of j; c,r are positive integers. The numbers in this sequence are sigma_1-numbers; those in A140480 are sigma_2-numbers. - Ctibor O. Zizka, Jul 14 2008 {a(n)} = union A175678 and A175679 where A175678 = numbers m such that the arithmetic mean Ad(m) of divisors of m and the arithmetic mean Ah(m) of numbers h < m such that gcd(h,m) = 1 are both integers and A175679 = numbers m such that the arithmetic mean Ad(m) of the divisors of m and the arithmetic mean Ak(m) of the numbers k <= m are both integers. - Jaroslav Krizek, Aug 07 2010
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, B2.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.51.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, The Biharmonic mean, arXiv:1601.03081 [math.NT], 2016. Paul T. Bateman, Paul Erdős, Carl Pomerance, and E. G. Straus, The arithmetic mean of the divisors of an integer (1981). In Knopp, M.I. ed., Analytic number theory, Proc. Conf., Temple Univ., 1980. Lecture Notes in Mathematics. 899. Springer-Verlag., pp. 197-220.
EXAMPLE
Sigma(6) = 12, tau(6) = 4, sigma(6)/tau(6) = 3 so 6 belongs to this sequence. - Bernard Schott, Jun 07 2017
MAPLE
with(numtheory); t := [ ]: f := [ ]: for n from 1 to 500 do if sigma(n) mod tau(n) = 0 then t := [ op(t), n ] else f := [ op(f), n ]; fi; od: t; # corrected by Wesley Ivan Hurt, Oct 03 2013
MATHEMATICA
Select[Range[120], IntegerQ[DivisorSigma[1, # ]/DivisorSigma[0, # ]] &] (* Stefan Steinerberger, Apr 03 2006 *)
PROG
(Haskell)
a003601 n = a003601_list !! (n-1)
a003601_list = filter ((== 1) . a245656) [1..]
(Python)
from sympy import divisors, divisor_count
[n for n in range(1, 10**5) if not sum(divisors(n)) % divisor_count(n)] # Chai Wah Wu, Aug 05 2014 (GAP) a:=Filtered([1..110], n->Sigma(n) mod Tau(n)=0);; Print(a); # Muniru A Asiru, Jan 25 2019
Harmonic or Ore numbers: numbers k such that the harmonic mean of the divisors of k is an integer.
(Formerly M4185 N1743)
+10
114
1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190, 18600, 18620, 27846, 30240, 32760, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 332640, 360360, 539400, 695520, 726180, 753480, 950976, 1089270, 1421280, 1539720
COMMENTS
Note that the harmonic mean of the divisors of k = k*tau(k)/sigma(k).
Equivalently, k*tau(k)/sigma(k) is an integer, where tau(k) ( A000005) is the number of divisors of k and sigma(k) is the sum of the divisors of k ( A000203). Equivalently, the average of the divisors of k divides k.
Note that the average of the divisors of k is not necessarily an integer, so the above wording should be clarified as follows: k divided by the average is an integer. See A007340. - Thomas Ordowski, Oct 26 2014 Ore showed that every perfect number ( A000396) is harmonic. The converse does not hold: 140 is harmonic but not perfect. Ore conjectured that 1 is the only odd harmonic number. Other examples of power mean numbers k such that some power mean of the divisors of k is an integer are the RMS numbers A140480. - Ctibor O. Zizka, Sep 20 2008 Conjecture: Every harmonic number is practical ( A005153). I've verified this refinement of Ore's conjecture for all terms less than 10^14. - Jaycob Coleman, Oct 12 2013 Conjecture: All terms > 1 are Zumkeller numbers ( A083207). Verified for all n <= 50. - Ivan N. Ianakiev, Nov 22 2017 Kanold (1957) proved that the asymptotic density of the harmonic numbers is 0. - Amiram Eldar, Jun 01 2020 Zachariou and Zachariou (1972) called these numbers "Ore numbers", after the Norwegian mathematician Øystein Ore (1899 - 1968), who was the first to study them. Ore (1948) and Garcia (1954) referred to them as "numbers with integral harmonic mean of divisors". The term "harmonic numbers" was used by Pomerance (1973). They are sometimes called "harmonic divisor numbers", or "Ore's harmonic numbers", to differentiate them from the partial sums of the harmonic series. - Amiram Eldar, Dec 04 2020 Conjecture: all terms > 1 have a Mersenne prime as a factor. - Ivan Borysiuk, Jan 28 2024
REFERENCES
G. L. Cohen and Deng Moujie, On a generalization of Ore's harmonic numbers, Nieuw Arch. Wisk. (4), 16 (1998) 161-172.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, Section B2, pp. 74-75.
W. H. Mills, On a conjecture of Ore, Proc. Number Theory Conf., Boulder CO, 1972, 142-146.
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).
LINKS
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, The Biharmonic mean, arXiv:1601.03081 [math.NT], 2016. Graeme L. Cohen and Ronald M. Sorli, Harmonic seeds, Fibonacci Quart., Vol. 36, No. 5 (1998), pp. 386-390; errata, 39 (2001) 4. Carl Pomerance, On a Problem of Ore: Harmonic Numbers, unpublished manuscript, 1973; abstract *709-A5, Notices of the American Mathematical Society, Vol. 20, 1973, page A-648, entire volume.
EXAMPLE
k=140 has sigma_0(140)=12 divisors with sigma_1(140)=336. The average divisor is 336/12=28, an integer, and divides k: k=5*28, so 140 is in the sequence.
k=496 has sigma_0(496)=10, sigma_1(496)=992: the average divisor 99.2 is not an integer, but k/(sigma_1/sigma_0)=496/99.2=5 is an integer, so 496 is in the sequence.
MAPLE
q:= (p, k) -> p^k*(p-1)*(k+1)/(p^(k+1)-1):
filter:= proc(n) local t; mul(q(op(t)), t=ifactors(n)[2])::integer end proc:
MATHEMATICA
Do[ If[ IntegerQ[ n*DivisorSigma[0, n]/ DivisorSigma[1, n]], Print[n]], {n, 1, 1550000}]
Select[Range[1600000], IntegerQ[HarmonicMean[Divisors[#]]]&] (* Harvey P. Dale, Oct 20 2012 *)
PROG
(PARI) a(n)=if(n<0, 0, n=a(n-1); until(0==(sigma(n, 0)*n)%sigma(n, 1), n++); n) /* Michael Somos, Feb 06 2004 */ (Haskell)
import Data.Ratio (denominator)
import Data.List (genericLength)
a001599 n = a001599_list !! (n-1)
a001599_list = filter ((== 1) . denominator . hm) [1..] where
hm x = genericLength ds * recip (sum $ map (recip . fromIntegral) ds)
where ds = a027750_row x
(GAP) Concatenation([1], Filtered([2, 4..2000000], n->IsInt(n*Tau(n)/Sigma(n)))); # Muniru A Asiru, Nov 26 2018 (Python)
from sympy import divisor_sigma as sigma
def ok(n): return (n*sigma(n, 0))%sigma(n, 1) == 0
(Python)
from itertools import count, islice
from functools import reduce
from math import prod
from sympy import factorint
def A001599_gen(startvalue=1): # generator of terms >= startvalue for n in count(max(startvalue, 1)):
f = factorint(n)
s = prod((p**(e+1)-1)//(p-1) for p, e in f.items())
if not reduce(lambda x, y:x*y%s, (e+1 for e in f.values()), 1)*n%s:
yield n
CROSSREFS
See A003601 for analogs referring to arithmetic mean and A000290 for geometric mean of divisors. sigma_0(n) (or tau(n)) is the number of divisors of n ( A000005). sigma_1(n) (or sigma(n)) is the sum of the divisors of n ( A000203).
Least k such that H(k) > n, where H(k) is the harmonic number Sum_{i=1..k} 1/i.
(Formerly M1249 N1385)
+10
82
1, 2, 4, 11, 31, 83, 227, 616, 1674, 4550, 12367, 33617, 91380, 248397, 675214, 1835421, 4989191, 13562027, 36865412, 100210581, 272400600, 740461601, 2012783315, 5471312310, 14872568831, 40427833596, 109894245429, 298723530401, 812014744422
COMMENTS
For k >= 1, log(k + 1/2) + gamma < H(k) < log(k + 1/2) + gamma + 1/(24k^2), where gamma is Euler's constant ( A001620). It is likely that the upper and lower bounds have the same floor for all k >= 2, in which case a(n) = floor(exp(n-gamma) + 1/2) for all n >= 0. This remark is based on a simple heuristic argument. The lower and upper bounds differ by 1/(24k^2), so the probability that there's an integer between the two bounds is 1/(24k^2). Summing that over all k >= 2 gives the expected number of values of k for which there's an integer between the bounds. That sum equals Pi^2/144 - 1/24 ~ 0.02687. That's much less than 1, so it is unlikely that there are any such values of k.
(End)
Referring to A118050 and A118051, using a few terms of the asymptotic series for the inverse of H(x), we can get an expression which, with greater likelihood than mentioned above, should give a(n) for all n >= 0. For example, using the same type of heuristic argument given by Dean Hickerson, it can be shown that, with probability > 99.995%, we should have, for all n >= 0, a(n) = floor(u + 1/2 - 1/(24u) + 3/(640u^3)) where u = e^(n - gamma). - David W. Cantrell (DWCantrell(AT)sigmaxi.net) For k > 1, H(k) is never an integer. Hence apart from the first two terms this sequence coincides with A004080. - Nick Hobson, Nov 25 2006
REFERENCES
John H. Conway and R. K. Guy, "The Book of Numbers," Copernicus, an imprint of Springer-Verlag, NY, 1996, pages 258-259.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 83, p. 28, Ellipses, Paris 2008.
Ronald Lewis Graham, Donald Ervin Knuth and Oren Patashnik, "Concrete Mathematics, a Foundation for Computer Science," Addison-Wesley Publishing Co., Reading, MA, 1989, Page 258-264, 438.
H. P. Robinson, Letter to N. J. A. Sloane, Oct 23 1973.
W. Sierpiński, Sur les decompositions de nombres rationnels, Oeuvres Choisies, Académie Polonaise des Sciences, Warsaw, Poland, 1974, p. 181.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane, Illustration for sequence M4299 (= A007340) in The Encyclopedia of Integer Sequences (with Simon Plouffe), Academic Press, 1995. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Stewart, L'univers des nombres, pp. 54, Belin-Pour La Science, Paris 2000.
FORMULA
Note that the conditionally convergent series Sum_{k>=1} (-1)^(k+1)/k = log 2 ( A002162). It is conjectured that, for n > 1, a(n) = floor(exp(n-gamma) + 1/2). - Benoit Cloitre, Oct 23 2002
MATHEMATICA
fh[0]=0; fh[1]=1; fh[k_] := Module[{tmp}, If[Floor[tmp=Log[k+1/2]+EulerGamma]==Floor[tmp+1/(24k^2)], Floor[tmp], UNKNOWN]]; a[0]=1; a[1]=2; a[n_] := Module[{val}, val=Round[Exp[n-EulerGamma]]; If[fh[val]==n&&fh[val-1]==n-1, val, UNKNOWN]]; (* fh[k] is either floor(H(k)) or UNKNOWN *)
PROG
(PARI) a(n)=if(n, my(k=exp(n-Euler)); ceil(solve(x=k-1.5, k+.5, intnum(y=0, 1, (1-y^x)/(1-y))-n)), 1) \\ Charles R Greathouse IV, Jun 13 2012 (Haskell)
a002387 n = a002387_list !! n
a002387_list = f 0 1 where
f x k = if hs !! k > fromIntegral x
then k : f (x + 1) (k + 1) else f x (k + 1)
where hs = scanl (+) 0 $ map recip [1..]
CROSSREFS
Apart from initial terms, same as A004080.
Least k such that H(k) >= n, where H(k) is the harmonic number Sum_{i=1..k} 1/i.
+10
32
0, 1, 4, 11, 31, 83, 227, 616, 1674, 4550, 12367, 33617, 91380, 248397, 675214, 1835421, 4989191, 13562027, 36865412, 100210581, 272400600, 740461601, 2012783315, 5471312310, 14872568831, 40427833596, 109894245429, 298723530401, 812014744422
REFERENCES
Bruno Rizzi and Cristina Scagliarini: I numeri armonici. Periodico di matematiche, "Mathesis", pp. 17-58, 1986, numbers 1-2. [From Vincenzo Librandi, Jan 05 2009] W. Sierpiński, Sur les décompositions de nombres rationnels, Oeuvres Choisies, Académie Polonaise des Sciences, Warsaw, Poland, 1974, p. 181.
N. J. A. Sloane, Illustration for sequence M4299 (= A007340) in The Encyclopedia of Integer Sequences (with Simon Plouffe), Academic Press, 1995.
EXAMPLE
a(2)=4 because 1/1 + 1/2 + 1/3 + 1/4 > 2.
MATHEMATICA
aux[0] = 0; Do[aux[n] = Floor[Floor[Sum[1/i, {i, n}]]]; If[aux[n] > aux[n - 1], Print[n]], {n, 1, 14000}] (* José María Grau Ribas, Feb 20 2010 *) a[0] = 0; a[1] = 1; a[n_] := k /. FindRoot[ HarmonicNumber[k] == n, {k, Exp[n - EulerGamma]}, WorkingPrecision -> 50] // Ceiling; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Aug 13 2013, after Charles R Greathouse IV *)
PROG
(PARI) my(t=0, n=0); for(i=0, 10^20, if (i, t+=1./i); if(t>=n, print1(i, ", "); n++)) \\ Thomas Gettys (tpgettys(AT)comcast.net), Jan 21 2007; corrected by Michel Marcus, Jan 19 2022 (Haskell)
import Data.List (findIndex); import Data.Maybe (fromJust)
a004080 n = fromJust $
findIndex (fromIntegral n <=) $ scanl (+) 0 $ map recip [1..]
CROSSREFS
Apart from first two terms, same as A002387.
EXTENSIONS
a(27) from Thomas Gettys (tpgettys(AT)comcast.net), Dec 05 2006
a(28) from Thomas Gettys (tpgettys(AT)comcast.net), Jan 21 2007
Harmonic means of divisors of harmonic numbers.
(Formerly M0609 N0220)
+10
18
1, 2, 3, 5, 6, 5, 8, 9, 11, 10, 7, 15, 15, 14, 17, 24, 24, 21, 13, 19, 27, 25, 29, 26, 44, 44, 29, 46, 39, 46, 27, 42, 47, 47, 54, 35, 41, 60, 51, 37, 48, 45, 49, 50, 49, 53, 77, 86, 86, 51, 96, 75, 70, 80, 99, 110, 81, 84, 13, 102, 82, 96, 114, 53, 108, 115, 105, 116, 91, 85, 105
COMMENTS
Values of n*tau(n)/sigma(n) corresponding to terms of A001599, where tau(n) ( A000005) is the number of divisors of n and sigma(n) is the sum of the divisors of n ( A000203). Kanold (1957) proved that each term appears only a finite number of times. - Amiram Eldar, Jun 01 2020
REFERENCES
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).
LINKS
Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, The Biharmonic mean, arXiv:1601.03081 [math.NT], 2016.
MATHEMATICA
A001600 = Reap[Do[tau = DivisorSigma[0, n]; sigma = DivisorSigma[1, n]; h = n*tau/sigma; If[IntegerQ[h], Print[h]; Sow[h]], {n, 1, 90000000}]][[2, 1]](* Jean-François Alcover, May 11 2012 *)
PROG
(Haskell)
a001600 n = a001600_list !! (n-1)
a001600_list =
[numerator m | x <- [1..], let m = hm x, denominator m == 1] where
hm x = genericLength divs * recip (sum $ map recip divs)
where divs = map fromIntegral $ a027750_row x
(PARI) lista(nn) = for (n=1, nn, if (denominator(q=n*numdiv(n)/sigma(n)) == 1, print1(q, ", "))); \\ Michel Marcus, Jan 13 2016
1, 140, 270, 672, 1638, 2970, 6200, 8190, 18600, 18620, 27846, 30240, 32760, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 332640, 360360, 539400, 695520, 726180, 753480, 950976, 1089270, 1421280, 1539720
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, B2.
EXAMPLE
A001599(4) = 140, but 336 = sigma(140) <> 2*140 = 280. Thus, 140 is a harmonic number which is not perfect. - Muniru A Asiru, Nov 26 2018
MATHEMATICA
Select[Range[2 10^7], IntegerQ[HarmonicMean[Divisors[#]]] && !DivisorSigma[1, #]==2 # &] (* Vincenzo Librandi, Nov 27 2018 *)
PROG
(GAP) Concatenation([1], Filtered([2, 4..2000000], n->Sigma(n)<>2*n and IsInt(n*Tau(n)/Sigma(n)))); # Muniru A Asiru, Nov 26 2018 (PARI) isok(n) = my(sn = sigma(n)); (frac(n*numdiv(n)/sn) == 0) && (sn != 2*n); \\ Michel Marcus, Nov 28 2018
CROSSREFS
For the associated harmonic means, see A102408.
Minimum number of identical bricks of length 1 which, when stacked without mortar in the naive way, form a stack of length >=n.
+10
3
1, 5, 32, 228, 1675, 12368, 91381, 675215, 4989192, 36865413, 272400601, 2012783316, 14872568832, 109894245430, 812014744423, 6000022499694, 44334502845081, 327590128640501, 2420581837980562, 17885814992891027
COMMENTS
Note that one can do "better" in terms of projections if one groups the bricks asymmetrically into lozenges with holes. See the Ainsley and Drummond references. Ainsley considers only the case of four bricks, but achieves an overhang of (15 - 4*sqrt(2))/8, compared with 25/24 for the harmonic pile. - D. G. Rogers, Aug 31 2005
REFERENCES
N. J. A. Sloane, Illustration for sequence M4299 (= A007340) in The Encyclopedia of Integer Sequences (with Simon Plouffe), Academic Press, 1995.
EXAMPLE
Obviously a(1)=1. If the center of gravity of one brick is placed at the end of a second brick, the length of the stack of 2 bricks is 1.5. If the c.g. of that stack is placed at the end of a third brick, the length of the stack is 1.75. Continuing, we get a stack of length 1.916666... for 4 bricks and a stack of length 2.0416666... for 5 bricks. Thus a(2)=5.
MATHEMATICA
f[n_] := k /. FindRoot[HarmonicNumber[k -1] == 2n, {k, Exp[ 2n]}, WorkingPrecision -> 100] // Ceiling; Array[f, 21, 0] (* Robert G. Wilson v, Jan 26 2017 after Jean-François Alcover in A014537 *) (* note that the index is off by one *)
Unitary arithmetic numbers k whose mean unitary divisor is a unitary divisor of k.
+10
3
1, 6, 60, 420, 630, 5460, 8190, 16632, 64260, 143640, 172900, 598500, 716625, 790398, 791700, 1182384, 1187550, 1530144, 2708160, 4277448, 5314680, 6284250, 6397300, 6741630, 14619150, 15214500, 22144500, 24315984, 87966648, 93284100, 161670600, 165197760, 232517250
COMMENTS
Also, unitary harmonic numbers k whose harmonic mean of the unitary divisors of k is a unitary divisor of k.
EXAMPLE
6 is a term since the arithmetic mean of its unitary divisors, {1, 2, 3, 6}, is 3, and 3 is also a unitary divisor of 6.
MATHEMATICA
q[n_] := Module[{f = FactorInteger[n], d, s, m}, d = 2^Length[f]; s = Times @@ (1 + Power @@@ f); m = s/d; IntegerQ[m] && Divisible[n, m] && CoprimeQ[m, n/m]]; Select[Range[10^6], q]
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