A050999 -id:A050999 - cp's OEIS frontend

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

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)

A320901 Expansion of Sum_{k>=1} x^k/(1 + x^k)^4.

Original entry on oeis.org

1, -3, 11, -23, 36, -49, 85, -143, 176, -188, 287, -433, 456, -479, 726, -959, 970, -1024, 1331, -1748, 1866, -1741, 2301, -3153, 2961, -2824, 3830, -4559, 4496, -4514, 5457, -6943, 6842, -6174, 7890, -9844, 9140, -8553, 11126, -13348, 12342, -11998, 14191, -16941

Offset: 1

Ilya Gutkovskiy, Oct 23 2018

sign

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

Cf. A000203, A000292, A000593, A001157, A001158, A002129, A050999, A051000, A059358, A138503, A320900, A321543.

Cf. A363598, A363616, A363617, A363631.

seq(coeff(series(add(x^k/(1+x^k)^4,k=1..n),x,n+1), x, n), n = 1 .. 45); # Muniru A Asiru, Oct 23 2018

nmax = 44; Rest[CoefficientList[Series[Sum[x^k/(1 + x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[(-1)^(d + 1) d (d + 1) (d + 2)/6, {d, Divisors[n]}], {n, 44}]

a(n) = sumdiv(n, d, (-1)^(d+1)*d*(d + 1)*(d + 2)/6); \\ Amiram Eldar, Jan 04 2025

G.f.: Sum_{k>=1} (-1)^(k+1)*A000292(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^(d+1)*d*(d + 1)*(d + 2)/6.
a(n) = (4*A000593(n) + 6*A050999(n) + 2*A051000(n) - 2*A000203(n) - 3*A001157(n) - A001158(n))/6.
a(n) = (A138503(n) + 3*A321543(n) + 2*A002129(n)) / 6. - Amiram Eldar, Jan 04 2025

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)

A347161 Sum of squares of odd divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 10, 1, 1, 10, 1, 1, 10, 1, 1, 10, 1, 1, 10, 1, 1, 35, 1, 1, 10, 1, 26, 10, 1, 1, 10, 26, 1, 10, 1, 1, 35, 1, 1, 10, 1, 26, 10, 1, 1, 10, 26, 50, 10, 1, 1, 35, 1, 1, 59, 1, 26, 10, 1, 1, 10, 75, 1, 10, 1, 1, 35, 1, 50, 10, 1, 26

Offset: 1

Ilya Gutkovskiy, Aug 20 2021

nonn

Cf. A001157, A050999, A333805, A333807, A339353, A347162.

Table[DivisorSum[n, #^2 &, # < Sqrt[n] && OddQ[#] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[(2 k - 1)^2 x^(2 k (2 k - 1))/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = my(r=sqrt(n)); sumdiv(n, d, if ((d%2) && (dMichel Marcus, Aug 21 2021

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

A347173 Sum of squares of odd divisors of n that are <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 10, 1, 1, 10, 1, 1, 10, 1, 1, 10, 1, 1, 10, 26, 1, 10, 1, 1, 35, 1, 1, 10, 1, 26, 10, 1, 1, 10, 26, 1, 10, 1, 1, 35, 1, 1, 10, 50, 26, 10, 1, 1, 10, 26, 50, 10, 1, 1, 35, 1, 1, 59, 1, 26, 10, 1, 1, 10, 75, 1, 10, 1, 1, 35, 1, 50, 10, 1, 26

Offset: 1

Ilya Gutkovskiy, Aug 21 2021

			a(18) = 10 as the odd divisors of 18 are the divisors of 9 which are 1, 3 and 9. Of those, 1 and 3 are <= sqrt(18) so we find the squares of 1 and 3 then add them i.e., a(18) = 1^2 + 3^2 = 10. - _David A. Corneth_, Feb 24 2024

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001157, A050999, A069288, A069289, A095118, A347161, A347174, A347175.

Table[DivisorSum[n, #^2 &, # <= Sqrt[n] && OddQ[#] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[(2 k - 1)^2 x^((2 k - 1)^2)/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sum(k=0, sqrtint(n), if ((k%2) && !(n%k), k^2)); \\ Michel Marcus, Aug 22 2021

a(n) = {
	my(s = sqrtint(n), res);
	n>>=valuation(n, 2);
	d = divisors(n);
	for(i = 1, #d,
		if(d[i] <= s,
			res += d[i]^2
		,
			return(res)
		)
	); res
} \\ David A. Corneth, Feb 24 2024

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

A076598 Sum of squares of divisors d of n such that d or n/d is odd.

Original entry on oeis.org

1, 5, 10, 17, 26, 50, 50, 65, 91, 130, 122, 170, 170, 250, 260, 257, 290, 455, 362, 442, 500, 610, 530, 650, 651, 850, 820, 850, 842, 1300, 962, 1025, 1220, 1450, 1300, 1547, 1370, 1810, 1700, 1690, 1682, 2500, 1850, 2074, 2366, 2650, 2210, 2570, 2451, 3255

Offset: 1

Vladeta Jovovic, Oct 20 2002

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

Cf. A002117, A069733, A069184, A001157, A050999, A076577.

f[2, e_] := 4^e+1 ; f[p_, e_] := (p^(2*e+2)-1)/(p^2-1) ; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 50] (* Amiram Eldar, Aug 01 2019 *)

a(n) = sumdiv(n, d, if ((d % 2) || (n/d % 2), d^2)); \\ Michel Marcus, Oct 30 2022

Multiplicative with a(2^e) = 4^e+1, a(p^e) = (p^(2*e+2)-1)/(p^2-1) for an odd prime p.
G.f.: Sum_{m>0} m^2*x^m*(1+2*x^m+3*x^(2*m))/(1+x^(2*m))/(1+x^m).
More generally, if b(n, k) is sum of k-th powers of divisors d of n such that d or n/d is odd then b(n, k) = sigma_k(n)-2^k*sigma_k(n/4) if n mod 4=0, otherwise b(n, k) = sigma_k(n).
G.f. for b(n, k): Sum_{m>0} m^k*x^m*(1+x^m+x^(2*m)-(2^k-1)*x^(3*m))/(1-x^(4*m)). b(n, k) is multiplicative and b(2^e, k) = 2^(k*e)+1, b(p^e, k) = (p^(k*e+k)-1)/(p^k-1) for an odd prime p.
a(n) = sigma_2(n)-4*sigma_2(n/4) if n mod 4=0, otherwise a(n) = sigma_2(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = 5*zeta(3)/16 = 0.375642... . - Amiram Eldar, Oct 30 2022

A363590 a(n) = Sum_{d|n, d odd} d^d.

Original entry on oeis.org

1, 1, 28, 1, 3126, 28, 823544, 1, 387420517, 3126, 285311670612, 28, 302875106592254, 823544, 437893890380862528, 1, 827240261886336764178, 387420517, 1978419655660313589123980, 3126, 5842587018385982521381947992, 285311670612

Offset: 1

Seiichi Manyama, Jul 08 2023

Not multiplicative: a(3)*a(5) != a(15), for example.

Cf. A062796, A333823, A333824.

Cf. A000593, A050999, A051000, A292919, A363991.

a[n_] := DivisorSum[n, #^# &, OddQ[#] &]; Array[a, 20] (* Amiram Eldar, Jul 26 2023 *)

PARI
```
a(n) = sumdiv(n, d, (d%2==1)*d^d);
```

from sympy import divisors
def A363590(n): return sum(d**d for d in divisors(n>>(~n & n-1).bit_length(),generator=True)) # Chai Wah Wu, Jul 09 2023

G.f.: Sum_{k>0} ((2*k-1) * x)^(2*k-1) / (1 - x^(2*k-1)).

a(2^n) = 1.

cp's OEIS Frontend

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

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A320901 Expansion of Sum_{k>=1} x^k/(1 + x^k)^4.

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

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

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

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

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A347161 Sum of squares of odd divisors of n that are < sqrt(n).

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A347173 Sum of squares of odd divisors of n that are <= sqrt(n).

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