A321810 -id:A321810 - cp's OEIS frontend

A050999 Sum of squares of odd divisors of n.

1, 1, 10, 1, 26, 10, 50, 1, 91, 26, 122, 10, 170, 50, 260, 1, 290, 91, 362, 26, 500, 122, 530, 10, 651, 170, 820, 50, 842, 260, 962, 1, 1220, 290, 1300, 91, 1370, 362, 1700, 26, 1682, 500, 1850, 122, 2366, 530, 2210, 10, 2451, 651, 2900, 170, 2810, 820, 3172, 50, 3620, 842, 3482

Offset: 1

Eric W. Weisstein

Denoted by Delta_2(n) in Glaisher 1907. - Michael Somos, May 17 2013

The sum of squares of even divisors of 2*k = 4*A001157(k), and the sum of squares of even divisors of 2*k-1 vanishes, for k >= 1. - Wolfdieter Lang, Jan 07 2017

			x + x^2 + 10*x^3 + x^4 + 26*x^5 + 10*x^6 + 50*x^7 + x^8 + 91*x^9 + 26*x^10 + ...

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).

Reinhard Zumkeller, 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).
R. J. Mathar, Survey of Dirichlet Series of Multiplicative Arithmetic Functions, arXiv:1106.4038 [math.NT], 2011, eq. (3.74).
Eric Weisstein's World of Mathematics, Odd Divisor Function.
Index entries for sequences mentioned by Glaisher

Cf. A051000 - A051002, A000593, A001227, A000203, A001157-A001160, A013954-A013972.

Glaisher's Delta_i (i=0..12): A001227, A000593, A050999, A051000, A051001, A051002, A321810, A321811, A321812, A321813, A321814, A321815, A321816

a050999 = sum . map (^ 2) . a182469_row
-- Reinhard Zumkeller, May 01 2012

a[n_] := 1/2*Sum[(1 - (-1)^d)*d^2, {d, Divisors[n]}]; Table[a[n], {n, 1, 59}] (* Jean-François Alcover, Oct 23 2012, from 2nd formula *)
a[ n_] := If[ n < 1, 0, Sum[ Mod[ d, 2] d^2, {d, Divisors@n}]] (* Michael Somos, May 17 2013 *)
f[p_, e_] := If[p == 2, 1, (p^(2*e + 2) - 1)/(p^2 - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 22 2020 *)
Table[Total[Select[Divisors[n],OddQ]^2],{n,80}] (* Harvey P. Dale, Jul 19 2024 *)

a(n)=sumdiv(n,d, if(d%2==1, d^2, 0 ) );  /* Joerg Arndt, Oct 07 2012 */

from sympy import divisor_sigma
def A050999(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),2)) # Chai Wah Wu, Jul 16 2022

From Vladeta Jovovic, Sep 10 2001: (Start)
Multiplicative with a(p^e) = 1 if p = 2, (p^(2e+2)-1)/(p^2-1) if p > 2.
a(n) = (1/2)*Sum_{d|n} (1-(-1)^d)*d^2.
a(2n) = sigma_2(2n) - 4*sigma_2(n), a(2n+1) = sigma_2(2n+1), where sigma_2(n) is sum of squares of divisors of n (A001157).
More generally, if b(n, k) is the sum of k-th powers of odd divisors of n then b(2n, k) = sigma_k(2n)-2^k*sigma_k(n), b(2n+1, k) = sigma_k(2n+1). b(n, k) is multiplicative with a(p^e) = 1 if p = 2, (p^(k*e+k)-1)/(p^k-1) if p > 2. (End)
G.f. for b(n, k): Sum_{m>0} m^k*x^m*(1-(2^k-1)*x^m)/(1-x^(2*m)). - Vladeta Jovovic, Oct 19 2002
Dirichlet g.f. (1-2^(2-s))*zeta(s)*zeta(s-2). - R. J. Mathar, Apr 06 2011
Dirichlet convolution of A001157 with [1,-4,0,0,0,0...]. Dirichlet convolution of [1,-3,1,-3,1,-3,..] with A000290. Dirichlet convolution of [1,0,9,0,25,0,49,0,81,...] with A000012 (or A057427). - R. J. Mathar, Jun 28 2011
a(n) = sum(A182469(n,k)^2: k=1..A001227(n)). [Reinhard Zumkeller, May 01 2012]
Sum_{k=1..n} a(k) ~ zeta(3) * n^3 / 6. - Vaclav Kotesovec, Nov 09 2018
G.f.: Sum_{n >= 1} x^n*(1 + 6*x^(2*n) + x^(4*n))/(1 - x^(2*n))^3. - Peter Bala, Dec 19 2021
Sum_{k=1..n} (-1)^(k+1) * a(k) ~ zeta(3) * n^3 / 8. - Vaclav Kotesovec, Aug 07 2022

A352034 Sum of the 6th powers of the odd proper divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 730, 1, 1, 730, 15626, 1, 730, 1, 117650, 16355, 1, 1, 532171, 1, 15626, 118379, 1771562, 1, 730, 15626, 4826810, 532171, 117650, 1, 11406980, 1, 1, 1772291, 24137570, 133275, 532171, 1, 47045882, 4827539, 15626, 1, 85884500, 1, 1771562, 11938421, 148035890

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 15626; a(10) = Sum_{d|10, d<10, d odd} d^6 = 1^6 + 5^6 = 15626.

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), A351647 (k=2), A352031 (k=3), A352032 (k=4), A352033 (k=5), this sequence (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).

Cf. A013665, A321810.

f[2, e_] := 1; f[p_, e_] := (p^(6*e+6) - 1)/(p^6 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^6, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)
Table[Total[Select[Most[Divisors[n]],OddQ]^6],{n,50}] (* Harvey P. Dale, Sep 15 2024 *)

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

G.f.: Sum_{k>=1} (2*k-1)^6 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022

For odd n >1, a(n) = A321810(n)-n^6; for even n, a(n) = A321810(n). - R. J. Mathar, Aug 15 2023

Sum_{k=1..n} a(k) ~ c * n^7, where c = (zeta(7)-1)/14 = 0.0005963769... . - Amiram Eldar, Oct 11 2023

A352052 Sum of the 6th powers of the divisor complements of the odd proper divisors of n.

Original entry on oeis.org

0, 64, 729, 4096, 15625, 46720, 117649, 262144, 532170, 1000064, 1771561, 2990080, 4826809, 7529600, 11406979, 16777216, 24137569, 34058944, 47045881, 64004096, 85884499, 113379968, 148035889, 191365120, 244156250, 308915840, 387952659, 481894400, 594823321

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 10^6 * Sum_{d|10, d<10, d odd} 1 / d^6 = 10^6 * (1/1^6 + 1/5^6) = 1000064.

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), A352048 (k=2), A352049 (k=3), A352050 (k=4), A352051 (k=5), this sequence (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Cf. A000035, A006519, A013665, A321810.

f:= proc(n) local m,d;
      m:= n/2^padic:-ordp(n,2);
      add((n/d)^6, d = select(`<`,numtheory:-divisors(m),n))
end proc:
map(f, [$1..30]); # Robert Israel, Apr 03 2023

Table[n^6*DivisorSum[n, 1/#^6 &, And[# < n, OddQ[#]] &], {n, 29}] (* Michael De Vlieger, Apr 04 2023 *)
a[n_] := DivisorSigma[-6, n/2^IntegerExponent[n, 2]] * n^6 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n^6*sumdiv(n, d, if ((dMichel Marcus, Apr 04 2023

a(n) = n^6 * sigma(n >> valuation(n, 2), -6) - n % 2; \\ Amiram Eldar, Oct 13 2023

a(n) = n^6 * Sum_{d|n, d

G.f.: Sum_{k>=2} k^6 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 18 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A321810(n) * A006519(n)^6 - A000035(n).

Sum_{k=1..n} a(k) = c * n^7 / 7, where c = 127*zeta(7)/128 = 1.000471548... . (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)

A321558 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^2.

Original entry on oeis.org

1, -5, 10, -13, 26, -50, 50, -45, 91, -130, 122, -130, 170, -250, 260, -173, 290, -455, 362, -338, 500, -610, 530, -450, 651, -850, 820, -650, 842, -1300, 962, -685, 1220, -1450, 1300, -1183, 1370, -1810, 1700, -1170, 1682, -2500, 1850, -1586, 2366

Offset: 1

N. J. A. Sloane, Nov 23 2018

			G.f. = x - 5*x^2 + 10*x^3 - 13*x^4 + 26*x^5 - 50*x^6 + 50*x^7 + ... - _Michael Somos_, Oct 24 2019

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from G. C. Greubel)
Peter Bala, A signed Dirichlet product of arithmetical functions
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

Column k=2 of A322083.
Glaisher's xi_i (i=0..12): A228441, A109506, A321558, A321559, A321560, A321561, A321562, A321563, A321564, A321565, A321807, A321808, A321809
Cf. A321543 - A321557, A321810 - A321836 for similar sequences.
Cf. A078306, A328667.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(-1)^(k+1)*k^2*x^k/(1 + x^k) : k in [1..2*m]]) )); // G. C. Greubel, Nov 28 2018

a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^2 &]; Array[a, 50] (* Amiram Eldar, Nov 27 2018 *)

apply( A321558(n)=sumdiv(n, d, (-1)^(n\d-d)*d^2), [1..30]) \\ M. F. Hasler, Nov 26 2018

s=(sum((-1)^(k+1)*k^2*x^k/(1 + x^k)  for k in (1..50))).series(x, 30); a = s.coefficients(x, sparse=False); a[1:] # G. C. Greubel, Nov 28 2018

G.f.: Sum_{k>=1} (-1)^(k+1)*k^2*x^k/(1 + x^k). - Ilya Gutkovskiy, Nov 27 2018
G.f.: Sum_{k>=1} (-1)^(k+1)*(x^k - x^(2*k))/(1 + x^k)^3. - Michael Somos, Oct 24 2019
a(n) = -(-1)^n A328667(n). a(2*n + 1) = A078306(2*n + 1). a(2*n) = A078306(2*n) - 8*A078306(n). - Michael Somos, Oct 24 2019
From Peter Bala, Jan 29 2022: (Start)
Multiplicative with a(2^k) = - (2^(2*k+1) + 7)/3 for k >= 1 and a(p^k) = (p^(2*k+2) - 1)/(p^2 - 1) for odd prime p.
n^2 = (-1)^(n+1)*Sum_{d divides n} A067856(n/d)*a(d). (End)

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)

A321812 Sum of 8th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 6562, 1, 390626, 6562, 5764802, 1, 43053283, 390626, 214358882, 6562, 815730722, 5764802, 2563287812, 1, 6975757442, 43053283, 16983563042, 390626, 37828630724, 214358882, 78310985282, 6562, 152588281251, 815730722, 282472589764

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=8 of A285425.
Cf. A050999, A051000, A051001, A051002, A321810 - A321816 (analog for 2nd .. 12th powers).
Cf. A321543 - A321565, A321807 - A321836 for related sequences.
Cf. A000265, A013667, A013956.

a[n_] := DivisorSum[n, #^8 &, OddQ[#] &]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)

apply( A321812(n)=sigma(n>>valuation(n,2),8), [1..30]) \\ M. F. Hasler, Nov 26 2018

from sympy import divisor_sigma
def A321812(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),8)) # Chai Wah Wu, Jul 16 2022

a(n) = A013956(A000265(n)) = sigma_8(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)^8*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^(8*e+8)-1)/(p^8-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^9, where c = zeta(9)/18 = 0.0556671... . (End)

A321813 Sum of 9th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 19684, 1, 1953126, 19684, 40353608, 1, 387440173, 1953126, 2357947692, 19684, 10604499374, 40353608, 38445332184, 1, 118587876498, 387440173, 322687697780, 1953126, 794320419872, 2357947692, 1801152661464, 19684, 3814699218751

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=9 of A285425.
Cf. A050999, A051000, A051001, A051002, A321810 - A321816 (analog for 2nd .. 12th powers).
Cf. A321543 - A321565, A321807 - A321836 for related sequences.
Cf. A000265, A013668, A013957.

a[n_] := DivisorSum[n, #^9 &, OddQ[#] &]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)

apply( A321813(n)=sigma(n>>valuation(n,2),9), [1..30]) \\ M. F. Hasler, Nov 26 2018

from sympy import divisor_sigma
def A321813(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),9)) # Chai Wah Wu, Jul 16 2022

a(n) = A013957(A000265(n)) = sigma_9(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)^9*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^(9*e+9)-1)/(p^9-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^10, where c = zeta(10)/20 = Pi^10/1871100 = 0.0500497... . (End)

A321814 Sum of 10th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 59050, 1, 9765626, 59050, 282475250, 1, 3486843451, 9765626, 25937424602, 59050, 137858491850, 282475250, 576660215300, 1, 2015993900450, 3486843451, 6131066257802, 9765626, 16680163512500, 25937424602, 41426511213650, 59050

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=10 of A285425.
Cf. A050999, A051000, A051001, A051002, A321810 - A321816 (analog for 2nd .. 12th powers).
Cf. A321543 - A321565, A321807 - A321836 for related sequences.
Cf. A000265, A013669, A013958.

a[n_] := DivisorSum[n, #^10 &, OddQ[#] &]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)

apply( A321814(n)=sigma(n>>valuation(n,2),10), [1..30]) \\ M. F. Hasler, Nov 26 2018

from sympy import divisor_sigma
def A321814(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),10)) # Chai Wah Wu, Jul 16 2022

a(n) = A013958(A000265(n)) = sigma_10(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)^10*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^(10*e+10)-1)/(p^10-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^11, where c = zeta(11)/22 = 0.045477... . (End)

A321815 Sum of 11th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 177148, 1, 48828126, 177148, 1977326744, 1, 31381236757, 48828126, 285311670612, 177148, 1792160394038, 1977326744, 8649804864648, 1, 34271896307634, 31381236757, 116490258898220, 48828126, 350279478046112, 285311670612, 952809757913928

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=11 of A285425.
Cf. A050999, A051000, A051001, A051002, A321810 - A321816 (analog for 2nd .. 12th powers).
Cf. A321543 - A321565, A321807 - A321836 for related sequences.
Cf. A000265, A013670, A013959.

List(List(List([1..25],j->DivisorsInt(j)),i->Filtered(i,k->IsOddInt(k))),m->Sum(m,n->n^11)); # Muniru A Asiru, Dec 07 2018

a[n_] := DivisorSum[n, #^11&, OddQ[#]&]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)

apply( A321815(n)=sigma(n>>valuation(n,2),11), [1..30]) \\ M. F. Hasler, Nov 26 2018

from sympy import divisor_sigma
def A321815(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),11)) # Chai Wah Wu, Jul 16 2022

a(n) = A013959(A000265(n)) = sigma_11(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)^11*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^(11*e+11)-1)/(p^11-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^12, where c = zeta(12)/24 = 691*Pi^12/15324309000 = 0.0416769... . (End)

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cp's OEIS Frontend

A050999 Sum of squares of odd divisors of n.

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A352034 Sum of the 6th powers of the odd proper divisors of n.

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A352052 Sum of the 6th powers of the divisor complements of the odd proper divisors of n.

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Maple

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

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A321558 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^2.

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

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A321812 Sum of 8th powers of odd divisors of n.

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