A050999 -id:A050999 - cp's OEIS frontend

A051002 Sum of 5th powers of odd divisors of n.

1, 1, 244, 1, 3126, 244, 16808, 1, 59293, 3126, 161052, 244, 371294, 16808, 762744, 1, 1419858, 59293, 2476100, 3126, 4101152, 161052, 6436344, 244, 9768751, 371294, 14408200, 16808, 20511150, 762744, 28629152, 1, 39296688, 1419858, 52541808, 59293, 69343958

Offset: 1

Eric W. Weisstein

The Apostol exercise F(x) is the g.f. of a(n)*(-1)^(n+1). - Michael Somos, Jul 05 2021

			G.f. = x + x^2 + 244*x^3 + x^4 + 3126*x^5 + 244*x^6 + 16808*x^7 + x^8 + ... - _Michael Somos_, Jul 05 2021

T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 25, Exercise 15.

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

Cf. A000593, A001227, A050999, A051000, A051001, A001160.

a[n_] := Select[ Divisors[n], OddQ]^5 // Total; Table[a[n], {n, 1, 34}] (* Jean-François Alcover, Oct 25 2012 *)
f[2, e_] := 1; f[p_, e_] := (p^(5*e + 5) - 1)/(p^5 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)
a[ n_] := If[n == 0, 0, DivisorSigma[5, n / 2^IntegerExponent[n, 2]]]; (* Michael Somos, Jul 05 2021 *)

a(n) = sumdiv(n , d, (d%2)*d^5); \\ Michel Marcus, Jan 14 2014

a(n)=sumdiv(n>>valuation(n,2), d, d^5) \\ Charles R Greathouse IV, Jul 05 2021

from sympy import divisor_sigma
def A051002(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),5)) # Chai Wah Wu, Jul 16 2022

Dirichlet g.f.: (1-2^(5-s))*zeta(s)*zeta(s-5). - R. J. Mathar, Apr 06 2011
G.f.: Sum_{k>=1} (2*k - 1)^5*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Jan 04 2017
The preceding g.f. is also 34*sigma_5(x^2) - 64*sigma_5(x^4) - sigma_5(-x), with sigma_5 the g.f. of A001160. Compare this with the Apostol reference which gives the g.f. of a(n)*(-1)^(n+1). - Wolfdieter Lang, Jan 31 2017
Multiplicative with a(2^e) = 1 and a(p^e) = (p^(5*e+5)-1)/(p^5-1) for p > 2. - Amiram Eldar, Sep 14 2020
Sum_{k=1..n} a(k) ~ Pi^6 * n^6 / 11340. - Vaclav Kotesovec, Sep 24 2020
G.f.: Sum_{n >= 1} x^n*R(5,x^(2*n))/(1 - x^(2*n))^6, where R(5,x) = 1 + 237*x + 1682*x^2 + 1682*x^3 + 237*x^4 + x^5 is the fifth row polynomial of A060187. - Peter Bala, Dec 20 2021

A285425 Square array A(n,k), n>=1, k>=0, read by antidiagonals, where column k is the expansion of Sum_{j>=1} (2j - 1)^kx^(2j-1)/(1 - x^(2j-1)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 10, 1, 2, 1, 1, 28, 1, 6, 2, 1, 1, 82, 1, 26, 4, 2, 1, 1, 244, 1, 126, 10, 8, 1, 1, 1, 730, 1, 626, 28, 50, 1, 3, 1, 1, 2188, 1, 3126, 82, 344, 1, 13, 2, 1, 1, 6562, 1, 15626, 244, 2402, 1, 91, 6, 2, 1, 1, 19684, 1, 78126, 730, 16808, 1, 757, 26, 12, 2

Offset: 1

Ilya Gutkovskiy, May 14 2017

A(n,k) is the sum of k-th powers of odd divisors of n.

			Square array begins:
1,  1,   1,    1,    1,     1,  ...
1,  1,   1,    1,    1,     1,  ...
2,  4,  10,   28,   82,   244,  ...
1,  1,   1,    1,    1,     1,  ...
2,  6,  26,  126,  626,  3126,  ...
2,  4,  10,   28,   82,   244,  ...

Eric Weisstein's World of Mathematics, Odd Divisor Function
Index entries for sequences related to sums of divisors

Columns k=0-5 give: A001227, A000593, A050999, A051000, A051001, A051002.

Cf. A109974, A279394, A292919.

Table[Function[k, SeriesCoefficient[Sum[(2 i - 1)^k x^(2 i - 1)/(1 - x^(2 i - 1)), {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 1, j}] // Flatten

G.f. of column k: Sum_{j>=1} (2*j - 1)^k*x^(2*j-1)/(1 - x^(2*j-1)).

Offset changed by Ilya Gutkovskiy, Oct 25 2018

A076577 Sum of squares of divisors d of n such that n/d is odd.

Original entry on oeis.org

1, 4, 10, 16, 26, 40, 50, 64, 91, 104, 122, 160, 170, 200, 260, 256, 290, 364, 362, 416, 500, 488, 530, 640, 651, 680, 820, 800, 842, 1040, 962, 1024, 1220, 1160, 1300, 1456, 1370, 1448, 1700, 1664, 1682, 2000, 1850, 1952, 2366, 2120, 2210, 2560, 2451, 2604

Offset: 1

Vladeta Jovovic, Oct 19 2002

			G.f. = x + 4*x^2 + 10*x^3 + 16*x^4 + 26*x^5 + 40*x^6 + 50*x^7 + 64*x^8 + ...

Robert Israel, 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).
Index entries for sequences mentioned by Glaisher

Cf. A001227, A002131, A001157, A050999.

Glaisher's Delta'_i (i=0..12): A001227, A002131, A076577, A007331, A285989, A096960, A321817, A096961, A321818, A096962, A321819, A096963, A321820

a:= n -> mul(`if`(t[1]=2, 2^(2*t[2]),
     (t[1]^(2*(1+t[2]))-1)/(t[1]^2-1)),t=ifactors(n)[2]):
map(a, [$1..100]); # Robert Israel, Jul 05 2016

a[ n_] := If[ n < 1, 0, Sum[ d^2 Mod[ n/d, 2], {d, Divisors @ n}]]; (* Michael Somos, Jun 09 2014 *)
Table[CoefficientList[Series[-Log[Product[(x^k - 1)^k/(x^k + 1)^k, {k, 1, 80}]]/2, {x, 0, 80}], x][[n + 1]] n, {n, 1, 80}] (* Benedict W. J. Irwin, Jul 05 2016 *)
f[2, e_] := 4^e; f[p_, e_] := (p^(2*e + 2) - 1)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

a(n) = sumdiv(n, d, d^2*((n/d) % 2)); \\ Michel Marcus, Jun 09 2014

G.f.: Sum_{m>0} m^2*x^m/(1-x^(2*m)). More generally, if b(n, k) is sum of k-th powers of divisors d of n such that n/d is odd then b(2n, k) = sigma_k(2n)-sigma_k(n), b(2n+1, k) = sigma_k(2n+1), where sigma_k(n) is sum of k-th powers of divisors of n. G.f. for b(n, k): Sum_{m>0} m^k*x^m/(1-x^(2*m)).
b(n, k) is multiplicative: b(2^e, k) = 2^(k*e), b(p^e, k) = (p^(ke+k)-1)/(p^k-1) for an odd prime p.
a(2*n) = sigma_2(2*n)-sigma_2(n), a(2*n+1) = sigma_2(2*n+1), where sigma_2(n) is sum of squares of divisors of n (cf. A001157).
b(n, k) = (sigma_k(2n)-sigma_k(n))/2^k. - Vladeta Jovovic, Oct 06 2003
Dirichlet g.f.: zeta(s)*(1-1/2^s)*zeta(s-2). - Geoffrey Critzer, Mar 28 2015
L.g.f.: -log(Product_{ k>0 } (x^k-1)^k/(x^k+1)^k)/2 = Sum_{ n>0 } (a(n)/n)*x^n. - Benedict W. J. Irwin, Jul 05 2016
Sum_{k=1..n} a(k) ~ 7*Zeta(3)*n^3 / 24. - Vaclav Kotesovec, Feb 08 2019

A285069 Expansion of Product_{k>=1} (1 - x^(2k-1))^(2k-1).

Original entry on oeis.org

1, -1, 0, -3, 3, -5, 8, -10, 22, -25, 41, -57, 88, -126, 168, -261, 351, -512, 685, -984, 1357, -1865, 2566, -3485, 4838, -6459, 8832, -11831, 16056, -21404, 28660, -38259, 50875, -67613, 89161, -118184, 155321, -204609, 267708, -351125, 458331, -597740

Offset: 0

Seiichi Manyama, Apr 09 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A050999, A262736, A262811.

CoefficientList[Series[Product[(1 - x^(2k-1))^(2k-1), {k, 50}], {x, 0, 50}], x] (* Indranil Ghosh, Apr 09 2017 *)

N=66; x='x+O('x^N); Vec(exp(-sum(k=1, N, sumdiv(k, d, d^2*(d%2))*x^k/k))) \\ Seiichi Manyama, Oct 31 2017

a(n) = (-1)^n * A262736(n).
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A050999(k)*a(n-k) for n > 0.
a(n) ~ (-1)^n * exp(3^(4/3) * (Zeta(3))^(1/3) * n^(2/3) / 2^(5/3)) * Zeta(3)^(1/6) / (2^(3/4) * 3^(1/3) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Nov 09 2017

A064027 a(n) = (-1)^nSum_{d|n} (-1)^dd^2.

Original entry on oeis.org

1, 3, 10, 19, 26, 30, 50, 83, 91, 78, 122, 190, 170, 150, 260, 339, 290, 273, 362, 494, 500, 366, 530, 830, 651, 510, 820, 950, 842, 780, 962, 1363, 1220, 870, 1300, 1729, 1370, 1086, 1700, 2158, 1682, 1500, 1850, 2318, 2366, 1590, 2210, 3390, 2451, 1953

Offset: 1

Vladeta Jovovic, Sep 11 2001

			L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 19*x^4/4 + 26*x^5/5 + 30*x^6/6 + ...
where exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 18*x^5 + 32*x^6 + 59*x^7 + 106*x^8 + 181*x^9 + ... + A224364(n)*x^n + ... - _Paul D. Hanna_, Apr 04 2013

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.

Paul D. Hanna, Table of n, a(n) for n = 1..1000
Heekyoung Hahn, Convolution sums of some functions on divisors, arXiv:1507.04426 [math.NT], 2015.

Cf. A001157, A002129, A008457, A050999, A224364.

Cf. A321543 - A321565, A321807 - A321836 for related sequences.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[k^2*x^k/(1-(-x)^k): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018

a[n_] := (-1)^n DivisorSum[n, (-1)^# * #^2 &]; Array[a, 50] (* Jean-François Alcover, Dec 23 2015 *)
a[n_] := If[OddQ[n], 1, (1 - 6/(4^(IntegerExponent[n, 2] + 1) - 1))] * DivisorSigma[2, n]; Array[a, 100] (* Amiram Eldar, Sep 21 2023 *)

{a(n)=if(n<1, 0, (-1)^n*sumdiv(n^1, d, (-1)^d*d^2))} \\ Paul D. Hanna, Apr 04 2013

Multiplicative with a(2^e) = (4^(e+1)-7)/3, a(p^e) = (p^(2*e+2)-1)/(p^2-1), p > 2.
a(n) = (-1)^n*(A001157(n) - 2*A050999(n)).
Logarithmic derivative of A224364. - Paul D. Hanna, Apr 04 2013
Bisection: a(2*k-1) = A001157(2*k-1), a(2*k) = 4*A001157(k) - A050999(2*k), k >= 1. In the Hardy reference a(n) = sigma^*2(n). - _Wolfdieter Lang, Jan 07 2017
G.f.: Sum_{k>=1} k^2*x^k/(1 - (-x)^k). - Ilya Gutkovskiy, Nov 09 2018
Sum_{k=1..n} a(k) ~ 7 * zeta(3) * n^3 / 24. - Vaclav Kotesovec, Nov 10 2018
Dirichlet g.f.: zeta(s) * zeta(s-2) * (1 - 1/2^(s-1) + 1/2^(2*s-3)). - Amiram Eldar, Sep 21 2023

A351647 Sum of the squares of the odd proper divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 10, 1, 1, 10, 26, 1, 10, 1, 50, 35, 1, 1, 91, 1, 26, 59, 122, 1, 10, 26, 170, 91, 50, 1, 260, 1, 1, 131, 290, 75, 91, 1, 362, 179, 26, 1, 500, 1, 122, 341, 530, 1, 10, 50, 651, 299, 170, 1, 820, 147, 50, 371, 842, 1, 260, 1, 962, 581, 1, 195, 1220, 1, 290

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 26; a(10) = Sum_{d|10, d<10, d odd} d^2 = 1^2 + 5^2 = 26.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of divisors.

Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), A091570 (k=1), this sequence (k=2), A352031 (k=3), A352032 (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).

Cf. A000035, A002117, A050999.

f[2, e_] := 1; f[p_, e_] := (p^(2*e+2) - 1)/(p^2 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^2, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)

a(n) = sumdiv(n, d, if ((d%2) && (dMichel Marcus, Mar 02 2022

a(n) = Sum_{d|n, d

G.f.: Sum_{k>=1} (2*k-1)^2 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022

From Amiram Eldar, Oct 11 2023: (Start)

a(n) = A050999(n) - n^2*A000035(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(3)-1)/6 = 0.0336761505... . (End)

A321810 Sum of 6th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 730, 1, 15626, 730, 117650, 1, 532171, 15626, 1771562, 730, 4826810, 117650, 11406980, 1, 24137570, 532171, 47045882, 15626, 85884500, 1771562, 148035890, 730, 244156251, 4826810, 387952660, 117650, 594823322, 11406980, 887503682

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=6 of A285425.
Cf. A050999, A051000, A051001, A051002, A321811 - A321816 (analog for 2nd .. 12th powers).
Cf. A321543 - A321565, A321807 - A321836 for related sequences.
Cf. A000265, A013665, A013954.

f[2, e_] := 1; f[p_, e_] := (p^(6*e + 6) - 1)/(p^6 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 02 2022 *)

apply( A321810(n)=sigma(n>>valuation(n,2),6), [1..30]) \\ M. F. Hasler, Nov 26 2018

from sympy import divisor_sigma
def A321810(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),6)) # Chai Wah Wu, Jul 16 2022

a(n) = A013954(A000265(n)) = sigma_6(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)^6*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 1 and a(p^e) = (p^(6*e+6)-1)/(p^6-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^7, where c = zeta(7)/14 = 0.0720249... . (End)
a(n) + a(n/2)*2^6 = A013954(n) where a(.)=0 for non-integer arguments. - R. J. Mathar, Aug 15 2023

A352048 Sum of the squares of the divisor complements of the odd proper divisors of n.

Original entry on oeis.org

0, 4, 9, 16, 25, 40, 49, 64, 90, 104, 121, 160, 169, 200, 259, 256, 289, 364, 361, 416, 499, 488, 529, 640, 650, 680, 819, 800, 841, 1040, 961, 1024, 1219, 1160, 1299, 1456, 1369, 1448, 1699, 1664, 1681, 2000, 1849, 1952, 2365, 2120, 2209, 2560, 2450, 2604, 2899, 2720

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 10^2 * Sum_{d|10, d<10, d odd} 1 / d^2 = 10^2 * (1/1^2 + 1/5^2) = 104.

Robert Israel, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), A352047 (k=1), this sequence (k=2), A352049 (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Cf. A006519, A050999, A076577, A233091.

f:= proc(n) local m,d;
      m:= n/2^padic:-ordp(n,2);
      add((n/d)^2, d = select(`<`,numtheory:-divisors(m),n))
end proc:
map(f, [$1..60]); # Robert Israel, Apr 03 2023

a[n_] := n^2 DivisorSum[n, If[# < n && OddQ[#], 1/#^2, 0]&];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, May 11 2023 *)
a[n_] := DivisorSigma[-2, n/2^IntegerExponent[n, 2]] * n^2 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n^2*sumdiv(n, d, if ((dMichel Marcus, May 11 2023

a(n) = n^2 * sigma(n >> valuation(n, 2), -2) - n % 2; \\ Amiram Eldar, Oct 13 2023

a(n) = n^2 * Sum_{d|n, d

G.f.: Sum_{k>=2} k^2 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 14 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A050999(n) * A006519(n)^2 - A000035(n).

Sum_{k=1..n} a(k) = c * n^3 / 3, where c = 7*zeta(3)/8 = 1.0517997... (A233091). (End)

A321816 Sum of 12th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 531442, 1, 244140626, 531442, 13841287202, 1, 282430067923, 244140626, 3138428376722, 531442, 23298085122482, 13841287202, 129746582562692, 1, 582622237229762, 282430067923, 2213314919066162, 244140626, 7355841353205284, 3138428376722

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=12 of A285425.
Cf. A050999, A051000, A051001, A051002, A321810 - A321815 (analog for 2nd .. 11th powers).
Cf. A321543 - A321565, A321807 - A321836 for related sequences.
Cf. A000265, A013671, A013960.

a[n_] := DivisorSum[n, #^12&, OddQ[#]&]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)

apply( A321816(n)=sigma(n>>valuation(n,2),12), [1..30]) \\ M. F. Hasler, Nov 26 2018

from sympy import divisor_sigma
def A321816(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),12)) # Chai Wah Wu, Jul 16 2022

a(n) = A013960(A000265(n)) = sigma_12(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)^12*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 1 and a(p^e) = (p^(12*e+12)-1)/(p^12-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^13, where c = zeta(13)/26 = 0.0384662... . (End)

A320900 Expansion of Sum_{k>=1} x^k/(1 + x^k)^3.

Original entry on oeis.org

1, -2, 7, -12, 16, -17, 29, -48, 52, -42, 67, -105, 92, -79, 142, -184, 154, -143, 191, -262, 266, -189, 277, -441, 341, -262, 430, -495, 436, -402, 497, -712, 634, -444, 674, -897, 704, -553, 878, -1118, 862, -766, 947, -1189, 1222, -807, 1129, -1753, 1254, -992

Offset: 1

Ilya Gutkovskiy, Oct 23 2018

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

Cf. A000203, A000217, A000593, A001157, A002129, A007437, A050999, A064027, A320901, A321543.

Cf. A363022, A363615, A363630.

seq(coeff(series(add(x^k/(1+x^k)^3,k=1..n),x,n+1), x, n), n = 1 .. 50); # Muniru A Asiru, Oct 23 2018

nmax = 50; Rest[CoefficientList[Series[Sum[x^k/(1 + x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[(-1)^(d + 1) d (d + 1)/2, {d, Divisors[n]}], {n, 50}]

a(n) = sumdiv(n, d, (-1)^(d+1)*d*(d + 1)/2); \\ Amiram Eldar, Jan 04 2025

G.f.: Sum_{k>=1} (-1)^(k+1)*A000217(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^(d+1)*d*(d + 1)/2.
a(n) = A000593(n) + A050999(n) - (A000203(n) + A001157(n))/2.
a(n) = (A002129(n) + A321543(n)) / 2. - Amiram Eldar, Jan 04 2025

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A051002 Sum of 5th powers of odd divisors of n.

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A285425 Square array A(n,k), n>=1, k>=0, read by antidiagonals, where column k is the expansion of Sum_{j>=1} (2*j - 1)^k*x^(2*j-1)/(1 - x^(2*j-1)).

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A076577 Sum of squares of divisors d of n such that n/d is odd.

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Maple

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A285069 Expansion of Product_{k>=1} (1 - x^(2*k-1))^(2*k-1).

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A064027 a(n) = (-1)^n*Sum_{d|n} (-1)^d*d^2.

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Magma

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A351647 Sum of the squares of the odd proper divisors of n.

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A321810 Sum of 6th powers of odd divisors of n.

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A285425 Square array A(n,k), n>=1, k>=0, read by antidiagonals, where column k is the expansion of Sum_{j>=1} (2j - 1)^kx^(2j-1)/(1 - x^(2j-1)).

A285069 Expansion of Product_{k>=1} (1 - x^(2k-1))^(2k-1).

A064027 a(n) = (-1)^nSum_{d|n} (-1)^dd^2.