A013955 -id:A013955 - cp's OEIS frontend

A282753 Expansion of phi_{9, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

0, 1, 516, 19692, 264208, 1953150, 10161072, 40353656, 135274560, 387597717, 1007825400, 2357947812, 5202783936, 10604499542, 20822486496, 38461429800, 69260574976, 118587876786, 200000421972, 322687698140, 516037855200, 794644193952, 1216701070992

Offset: 0

Seiichi Manyama, Feb 21 2017

Multiplicative because A013955 is. - Andrew Howroyd, Jul 25 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A282097 (phi_{3, 2}), A282099 (phi_{5, 2}), A282751 (phi_{7, 2}), this sequence (phi_{9, 2}).
Cf. A282752 (E_2^2*E_4^2), A282102 (E_2*E_4*E_6), A008411 (E_4^3), A280869 (E_6^2).
Cf. A013955 (sigma_7(n)), A282060 (n*sigma_7(n)), this sequence (n^2*sigma_7(n)).
Cf. A013666.

Table[If[n>0, n^2 * DivisorSigma[7, n], 0], {n, 0, 22}] (* Indranil Ghosh, Mar 12 2017 *)
nmax = 40; CoefficientList[Series[Sum[k^9*x^k*(1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 02 2025 *)

for(n=0, 22, print1(if(n==0, 0, n^2 * sigma(n, 7)),", ")) \\ Indranil Ghosh, Mar 12 2017

a(n) = n^2*A013955(n) for n > 0.
a(n) = (9*A282752(n) - 18*A282102(n) + 5*A008411(n) + 4*A280869(n))/8640.
Sum_{k=1..n} a(k) ~ zeta(8) * n^10 / 10. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^(2*e) * (p^(7*e+7)-1)/(p^7-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-9). (End)
G.f.: Sum_{k>=1} k^9*x^k*(1 + x^k)/(1 - x^k)^3. - Vaclav Kotesovec, Aug 02 2025

A301551 Expansion of Product_{k>=1} (1 + x^k)^(sigma_7(k)).

Original entry on oeis.org

1, 1, 129, 2317, 26957, 385147, 5514889, 70250881, 866874825, 10634404922, 126906497939, 1470673175003, 16705788322140, 186487470519166, 2044203433733016, 22025647881901542, 233686866722213324, 2443978994099801452, 25211475391206919299, 256716054713570158748

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A013955, A107742, A301545.

nmax = 30; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[7, k], {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(9 * Pi^(8/9) * (17*Zeta(9))^(1/9) * n^(8/9) / 2^(29/9)) * (17*Zeta(9)/Pi)^(1/18) / (3 * 2^(883/1440) * n^(5/9)).

G.f.: exp(Sum_{k>=1} sigma_8(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 26 2018

A372964 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( n/gcd(x_1, x_2, x_3, x_4, n) )^3.

Original entry on oeis.org

1, 121, 2161, 15481, 78001, 261481, 823201, 1981561, 4726081, 9438121, 19485841, 33454441, 62746321, 99607321, 168560161, 253639801, 410333761, 571855801, 893864881, 1207533481, 1778937361, 2357786761, 3404813281, 4282153321, 6093828001, 7592304841, 10335939121

Offset: 1

Seiichi Manyama, May 18 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A068970, A084220, A372950.
Cf. A008683, A013955.
Cf. A013663, A013666.
Cf. A350156, A372962.

f[p_, e_] := (p^(7*e+7) - p^(7*e+3) + p^3 - 1)/(p^7-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^3*sigma(d, 7));

a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_7(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(7*e+7) - p^(7*e+3) + p^3 - 1)/(p^7-1).
Dirichlet g.f.: zeta(s)*zeta(s-7)/zeta(s-3).
Sum_{k=1..n} a(k) ~ c * n^8 / 8, where c = zeta(8)/zeta(5) = 0.968319491... . (End)
a(n) = Sum_{d|n} phi(n/d) * (n/d)^6 * sigma_6(d^2)/sigma_3(d^2). - Seiichi Manyama, May 24 2024
a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, x_4, n) )^4. - Seiichi Manyama, May 25 2024

A055711 Numbers k such that k | sigma_7(k).

Original entry on oeis.org

1, 6, 28, 86, 120, 145, 258, 290, 435, 496, 580, 588, 672, 696, 870, 946, 1032, 1305, 1720, 1740, 2245, 2610, 2712, 2838, 3164, 3282, 3408, 3480, 3724, 3784, 4060, 4490, 5160, 5220, 6735, 6786, 6960, 7830, 8514, 8980, 9436, 9492, 9632, 9976

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

sigma_7(k) is the sum of the 7th powers of the divisors of k (A013955).

Problem 11090 proves that this sequence is infinite. - T. D. Noe, Apr 18 2006

Amiram Eldar, Table of n, a(n) for n = 1..10000
Florian Luca and John Ferdinands, Problem 11090: Sometimes n divides sigma_k(n), Amer. Math. Monthly 113:4 (2006), pp. 372-373.

Cf. A013955.

Cf. A055709, A055710, A055712, A055713.

Do[If[Mod[DivisorSigma[7, n], n]==0, Print[n]], {n, 1, 10000}]

is(n)=sigma(n,7)%n==0 \\ Charles R Greathouse IV, Feb 04 2013

A074868 Non-balanced numbers in A015765.

Original entry on oeis.org

295, 590, 767, 885, 1038, 1416, 1534, 1589, 1770, 2065, 2301, 2422, 3178, 3186, 3245, 3304, 3448, 3540, 4130, 4602, 4767, 5192, 5230, 5448, 5516, 5605, 6195, 6291, 6356, 6490, 6574, 6860, 7266, 7945, 7965, 8236, 8260, 8437, 8968, 9145, 9204, 9342

Offset: 1

Labos Elemer, Dec 05 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A013955, A015765, A020492, A078539.

Cf. A074866, A077801, A077803, A078544, A078549, A078550.

q[n_] := Sign[Mod[DivisorSigma[{1, 7}, n], EulerPhi[n]]] == {1, 0}; Select[Range[10000], q] (* Amiram Eldar, Apr 11 2024 *)

is(n) = {my(f = factor(n), phi = eulerphi(f)); (sigma(f) % phi) && !(sigma(f, 7) % phi);} \\ Amiram Eldar, Apr 11 2024

sigma_7(a(n)) mod phi(a(n)) = 0; sigma(a(n)) mod phi(a(n)) <> 0.

A211347 Numbers n such that n = sigma_k(m) for some k >= 1.

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 17, 18, 20, 21, 24, 26, 28, 30, 31, 32, 33, 36, 38, 39, 40, 42, 44, 48, 50, 54, 56, 57, 60, 62, 63, 65, 68, 72, 73, 74, 78, 80, 82, 84, 85, 90, 91, 93, 96, 98, 102, 104, 108, 110, 112, 114, 120, 121, 122

Offset: 1

Jon Perry, Feb 05 2013

nonn

Sigma_k(n) = Sum[d|n, d^k].
Sigma_0(n) can be any positive integer and so is ignored in this sequence.
The asymptotic density of this sequence is 0 (Niven, 1951, Rao and Murty, 1979). - Amiram Eldar, Jul 23 2020

			Sigma_2(4) = 1 + 4 + 16 = 21 so 21 is in the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Ivan Niven, The asymptotic density of sequences, Bull. Amer. Math. Soc., Vol. 57 (1951), pp. 420-434.
R. Sita Rama Chandra Rao and G. Sri Rama Chandra Murty, On a theorem of Niven, Canadian Mathematical Bulletin, Vol 22, No. 1 (1979), pp. 113-115.
Eric W. Weisstein, MathWorld: Divisor function

Cf. A000203, A001157, A001158, A001159, A001160.
Cf. A013954, A013955, A013956, A013957, A013958, A013959, A013960, A013961, A013962, A013963, A013964, A013965, A013966, A013967, A013968, A013969, A013970, A013971, A013972.
Cf. A000005.

upto[n_] := Select[Union@Flatten[{1, DivisorSigma[Range@Max[1,Floor@Log[#,n]], #] & /@ Range[2,n]}], # <= n &]; upto[122] (* Giovanni Resta, Feb 05 2013 *)

list(lim)=if(lim<3, return(if(lim<1,[],[1]))); my(v=List([1])); for(k=1,logint((lim\=1)-1,2), forfactored(m=2,sqrtnint(lim-1,k), my(t=sigma(m,k)); if(t<=lim, listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Apr 09 2022

A321811 Sum of 7th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 2188, 1, 78126, 2188, 823544, 1, 4785157, 78126, 19487172, 2188, 62748518, 823544, 170939688, 1, 410338674, 4785157, 893871740, 78126, 1801914272, 19487172, 3404825448, 2188, 6103593751, 62748518, 10465138360, 823544, 17249876310

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Eric Weisstein's World of Mathematics, Odd Divisor Function.
Index entries for sequences mentioned by Glaisher.

Column k=7 of A285425.
Cf. A050999, A051000, A051001, A051002, A321810 - A321816 (analog for 2nd .. 12th powers).
Cf. A321543 - A321565, A321807 - A321836 for related sequences.
Cf. A000265, A013666, A013955.

a[n_] := DivisorSum[n, #^7 &, OddQ[#] &]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)

apply( A321811(n)=sigma(n>>valuation(n,2),7), [1..30]) \\ M. F. Hasler, Nov 26 2018

from sympy import divisor_sigma
def A321811(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),7)) # Chai Wah Wu, Jul 16 2022

a(n) = A013955(A000265(n)) = sigma_7(odd part of n); in particular, a(2^k) = 1 for all k >= 0. - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (2*k - 1)^7*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 07 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 1 and a(p^e) = (p^(7*e+7)-1)/(p^7-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^8, where c = zeta(8)/16 = Pi^8/151200 = 0.0627548... . (End)

A365664 Expansion of Sum_{0

Original entry on oeis.org

1, 3, 9, 22, 51, 97, 188, 330, 568, 918, 1452, 2233, 3344, 4884, 7004, 9856, 13653, 18699, 25080, 33462, 43918, 57304, 73668, 94482, 119262, 150285, 187231, 232560, 285660, 350746, 425627, 516477, 620731, 745503, 887796, 1056669, 1247521, 1472460, 1726054, 2021327

Offset: 10

Seiichi Manyama, Sep 15 2023

nonn

Number of partitions of n with four designated summands. For example: a(11) = 3 because there are three partitions of 11 with four designated summands: [5'+ 3'+ 2'+ 1'], [4'+ 3'+ 2'+ 1'+ 1], [4'+ 3'+ 2'+ 1 + 1']. - Omar E. Pol, Jul 26 2025

Seiichi Manyama, Table of n, a(n) for n = 10..10000
George E. Andrews and Simon C. F. Rose, MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms, Journal für die reine und angewandte Mathematik (Crelles Journal), Vol. 2013, No. 676 (2013), pp. 97-103; arXiv preprint, arXiv:1010.5769 [math.NT], 2010.

A diagonal of A060043.
Cf. A000203, A002127, A002128, A365665.
Cf. A010816, A365630, A365666.
Cf. A001158, A001160, A013955.
Column k=4 of A385001.

a[n_] := Module[{d = DivisorSigma[{1, 3, 5, 7}, n]}, (5*d[[4]] - (126*n-441)*d[[3]] + (756*n^2-4410*n+4935)*d[[2]] - (840*n^3-5880*n^2+9870*n-3229)*d[[1]])/967680]; Array[a, 40, 10] (* Amiram Eldar, Jan 07 2025 *)

a(n) = (5*sigma(n, 7)-(126*n-441)*sigma(n, 5)+(756*n^2-4410*n+4935)*sigma(n, 3)-(840*n^3-5880*n^2+9870*n-3229)*sigma(n))/967680; \\ Seiichi Manyama, Jul 24 2024

G.f.: (1/9) * ( Sum_{k>=4} (-1)^k * (2*k+1) * binomial(k+4,8) * q^(k*(k+1)/2) ) / ( Sum_{k>=0} (-1)^k * (2*k+1) * q^(k*(k+1)/2) ).
a(n) = (5*sigma_7(n) - (126*n-441)*sigma_5(n) + (756*n^2-4410*n+4935)*sigma_3(n) - (840*n^3-5880*n^2+9870*n-3229)*sigma(n))/967680. - Seiichi Manyama, Jul 24 2024
Sum_{k=1..n} a(k) ~ Pi^8 * n^8 / (8!*9!). - Vaclav Kotesovec, Aug 01 2025

A386781 a(n) = n^3*sigma_7(n).

Original entry on oeis.org

0, 1, 1032, 59076, 1056832, 9765750, 60966432, 282475592, 1082196480, 3488379453, 10078254000, 25937425932, 62433407232, 137858494046, 291514810944, 576921447000, 1108169199616, 2015993905362, 3600007595496, 6131066264660, 10320757104000, 16687528072992, 26767423561824

Offset: 0

Vaclav Kotesovec, Aug 02 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A282211, A386746, A282213, A386748, A282781, A386780, A386782.
Cf. A013955, A282060, A282753.
Cf. A386813, A282549, A282792, A058550, A282576.

[0] cat [n^3*DivisorSigma(7, n): n in [1..35]]; // Vincenzo Librandi, Aug 04 2025

Table[n^3*DivisorSigma[7, n], {n, 0, 30}]
(* or *)
nmax = 30; CoefficientList[Series[Sum[k^10*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]
(* or *)
terms = 30; E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}]; E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}]; E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}]; CoefficientList[Series[(3*E2[x]^3*E4[x]^2 + 5*E2[x]*E4[x]^3 - 9*E2[x]^2*E4[x]*E6[x] - 3*E4[x]^2*E6[x] + 4*E2[x]*E6[x]^2)/3456, {x, 0, terms}], x]

G.f.: Sum_{k>=1} k^10*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4.
a(n) = (3*A386813(n) + 5*A282549(n) - 9*A282792(n) - 3*A058550(n) + 4*A282576(n))/3456.
a(n) = n^3*A013955(n).
Dirichlet g.f.: zeta(s-3)*zeta(s-10). - R. J. Mathar, Aug 03 2025

A076234 Numbers k such that sigma(k)/k, sigma_3(k)/k, sigma_5(k)/k and sigma_7(k)/k are all integers.

Original entry on oeis.org

1, 6, 120, 672, 30240, 32760, 33550336, 459818240, 1379454720, 8589869056, 31998395520, 51001180160, 137438691328, 153003540480, 30823866178560, 796928461056000, 6088728021160320, 212517062615531520, 2305843008139952128, 69357059049509038080, 143573364313605309726720

Offset: 1

Labos Elemer, Oct 04 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..55

Cf. A000203, A001158, A001160, A013955.
Cf. A007691, A046763, A055709, A076231, A076233.
Cf. A066289 (k divides sigma_m(k) for all odd m).

isok(n) = !(sigma(n) % n) && !(sigma(n, 3) % n) && !(sigma(n, 5) % n) && !(sigma(n, 7) % n); \\ Michel Marcus, Dec 26 2013

a(13)-a(18) from Donovan Johnson, May 08 2010

a(19)-a(21) from Amiram Eldar, May 09 2024

cp's OEIS Frontend

A282753 Expansion of phi_{9, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

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A301551 Expansion of Product_{k>=1} (1 + x^k)^(sigma_7(k)).

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A372964 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( n/gcd(x_1, x_2, x_3, x_4, n) )^3.

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A055711 Numbers k such that k | sigma_7(k).

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A074868 Non-balanced numbers in A015765.

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A211347 Numbers n such that n = sigma_k(m) for some k >= 1.

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A321811 Sum of 7th powers of odd divisors of n.

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A365664 Expansion of Sum_{0

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