A321543 -id:A321543 - cp's OEIS frontend

A321552 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^7.

1, 127, 2188, 16255, 78126, 277876, 823544, 2080639, 4785157, 9922002, 19487172, 35565940, 62748518, 104590088, 170939688, 266321791, 410338674, 607714939, 893871740, 1269938130, 1801914272, 2474870844, 3404825448, 4552438132, 6103593751, 7969061786, 10465138360, 13386707720, 17249876310

Offset: 1

N. J. A. Sloane, Nov 23 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).
Index entries for sequences mentioned by Glaisher.

Sum_{k>=1} k^b*x^k/(1 + x^k): A000593 (b=1), A078306 (b=2), A078307 (b=3), A284900 (b=4), A284926 (b=5), A284927 (b=6), this sequence (b=7), A321553 (b=8), A321554 (b=9), A321555 (b=10), A321556 (b=11), A321557 (b=12).
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A013666.

f[p_, e_] := (p^(7*e + 7) - 1)/(p^7 - 1); f[2, e_] := (63*2^(7*e + 1) + 1)/127; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)

apply( A321552(n)=sumdiv(n, d, (-1)^(n\d-1)*d^7), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^7*x^k/(1 + x^k). - Seiichi Manyama, Nov 23 2018
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (63*2^(7*e+1)+1)/127, and a(p^e) = (p^(7*e+7) - 1)/(p^7 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^8, where c = 127*zeta(8)/1024 = 0.124529... . (End)

A321557 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^12.

Original entry on oeis.org

1, 4095, 531442, 16773119, 244140626, 2176254990, 13841287202, 68702695423, 282430067923, 999755863470, 3138428376722, 8913939907598, 23298085122482, 56680071092190, 129746582562692, 281406240452607, 582622237229762, 1156551128144685, 2213314919066162, 4094999772632494

Offset: 1

N. J. A. Sloane, Nov 23 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).
Index entries for sequences mentioned by Glaisher.

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

Cf. A013671.

f[p_, e_] := (p^(12*e + 12) - 1)/(p^12 - 1); f[2, e_] := (2047*2^(12*e + 1) + 1)/4095; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)

apply( A321557(n)=sumdiv(n, d, (-1)^(n\d-1)*d^12), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^12*x^k/(1 + x^k). - Seiichi Manyama, Nov 25 2018
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (2047*2^(12*e+12)+1)/4095, and a(p^e) = (p^(12*e+12) - 1)/(p^12 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^13, where c = 315*zeta(13)/4096 = 0.0769137... . (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)

A363022 Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^3.

Original entry on oeis.org

0, 1, -3, 7, -10, 13, -21, 35, -39, 36, -55, 85, -78, 71, -118, 155, -136, 130, -171, 232, -234, 177, -253, 389, -310, 248, -390, 455, -406, 378, -465, 651, -586, 426, -626, 832, -666, 533, -822, 1040, -820, 734, -903, 1129, -1144, 783, -1081, 1637, -1197, 961, -1414, 1580, -1378

Offset: 1

Seiichi Manyama, Jun 11 2023

sign

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

Cf. A325940, A363598, A363613, A363614.

Cf. A002129, A069153, A321543.

a[n_] := DivisorSum[n, (-1)^# * Binomial[#, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

my(N=60, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1+x^k)^3)))

a(n) = sumdiv(n, d, (-1)^d*binomial(d, 2));

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

a(n) = Sum_{d|n} (-1)^d * binomial(d,2) = (A002129(n) - A321543(n))/2.

A321553 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^8.

Original entry on oeis.org

1, 255, 6562, 65279, 390626, 1673310, 5764802, 16711423, 43053283, 99609630, 214358882, 428360798, 815730722, 1470024510, 2563287812, 4278124287, 6975757442, 10978587165, 16983563042, 25499674654, 37828630724, 54661514910, 78310985282, 109660357726, 152588281251, 208011334110, 282472589764

Offset: 1

N. J. A. Sloane, Nov 23 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).
Index entries for sequences mentioned by Glaisher.

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

Cf. A013667.

Table[Total[(-1)^(n/#+1) #^8&/@Divisors[n]],{n,30}] (* Harvey P. Dale, May 05 2021 *)
f[p_, e_] := (p^(8*e + 8) - 1)/(p^8 - 1); f[2, e_] := (127*2^(8*e + 1) + 1)/255; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)

apply( A321553(n)=sumdiv(n, d, (-1)^(n\d-1)*d^8), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^8*x^k/(1 + x^k). - Seiichi Manyama, Nov 23 2018
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (127*2^(8*e+1)+1)/255, and a(p^e) = (p^(8*e+8) - 1)/(p^8 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^9, where c = 85*zeta(9)/768 = 0.110899... . (End)

A321554 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^9.

Original entry on oeis.org

1, 511, 19684, 261631, 1953126, 10058524, 40353608, 133955071, 387440173, 998047386, 2357947692, 5149944604, 10604499374, 20620693688, 38445332184, 68584996351, 118587876498, 197981928403, 322687697780, 510998308506, 794320419872, 1204911270612, 1801152661464, 2636771617564, 3814699218751

Offset: 1

N. J. A. Sloane, Nov 23 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).
Index entries for sequences mentioned by Glaisher.

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

Cf. A013668.

f[p_, e_] := (p^(9*e + 9) - 1)/(p^9 - 1); f[2, e_] := (255*2^(9*e + 1) + 1)/511; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)

apply( A321554(n)=sumdiv(n, d, (-1)^(n\d-1)*d^9), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^9*x^k/(1 + x^k). - Seiichi Manyama, Nov 24 2018
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (255*2^(9*e+1)+1)/511, and a(p^e) = (p^(9*e+9) - 1)/(p^9 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^10, where c = 511*zeta(10)/5120 = 0.0999039... . (End)

A321555 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^10.

Original entry on oeis.org

1, 1023, 59050, 1047551, 9765626, 60408150, 282475250, 1072692223, 3486843451, 9990235398, 25937424602, 61857886550, 137858491850, 288972180750, 576660215300, 1098436836351, 2015993900450, 3567040850373, 6131066257802, 10229991281926, 16680163512500, 26533985367846, 41426511213650, 63342475768150

Offset: 1

N. J. A. Sloane, Nov 23 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).
Index entries for sequences mentioned by Glaisher.

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

Cf. A013669.

f[p_, e_] := (p^(10*e + 10) - 1)/(p^10 - 1); f[2, e_] := (511*2^(10*e + 1) + 1)/1023; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)

apply( A321555(n)=sumdiv(n, d, (-1)^(n\d-1)*d^10), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^10*x^k/(1 + x^k). - Seiichi Manyama, Nov 25 2018
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (511*2^(10*e+1)+1)/1023, and a(p^e) = (p^(10*e+10) - 1)/(p^10 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^11, where c = 93*zeta(11)/1024 = 0.0908651... . (End)

A321556 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^11.

Original entry on oeis.org

1, 2047, 177148, 4192255, 48828126, 362621956, 1977326744, 8585738239, 31381236757, 99951173922, 285311670612, 742649588740, 1792160394038, 4047587844968, 8649804864648, 17583591913471, 34271896307634, 64237391641579

Offset: 1

N. J. A. Sloane, Nov 23 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).
Index entries for sequences mentioned by Glaisher.

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

Cf. A013670.

f[p_, e_] := (p^(11*e + 11) - 1)/(p^11 - 1); f[2, e_] := (1023*2^(11*e + 1) + 1)/2047; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)

apply( A321556(n)=sumdiv(n, d, (-1)^(n\d-1)*d^11), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^11*x^k/(1 + x^k). - Seiichi Manyama, Nov 25 2018
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (1023*2^(11*e+1)+1)/2047, and a(p^e) = (p^(11*e+11) - 1)/(p^11 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^12, where c = 2047*zeta(12)/24576 = 0.0833131... . (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)

A344300 Expansion of Sum_{k>=1} (-1)^(k+1) * k^2 * x^(k^2) / (1 - x^(k^2)).

Original entry on oeis.org

1, 1, 1, -3, 1, 1, 1, -3, 10, 1, 1, -3, 1, 1, 1, -19, 1, 10, 1, -3, 1, 1, 1, -3, 26, 1, 10, -3, 1, 1, 1, -19, 1, 1, 1, -30, 1, 1, 1, -3, 1, 1, 1, -3, 10, 1, 1, -19, 50, 26, 1, -3, 1, 10, 1, -3, 1, 1, 1, -3, 1, 1, 10, -83, 1, 1, 1, -3, 1, 1, 1, -30, 1, 1, 26, -3, 1, 1, 1, -19

Offset: 1

Ilya Gutkovskiy, May 14 2021

Excess of sum of odd squares dividing n over sum of even squares dividing n.

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

Cf. A002129, A035316, A300853, A321543, A344299.

nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) k^2 x^(k^2)/(1 - x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + 1) # &, IntegerQ[#^(1/2)] &], {n, 1, 80}]
f[p_, e_] := (p^(2*Floor[e/2] + 2) - 1)/(p^2 - 1); f[2, e_] := 2 - (2^(2*Floor[e/2] + 2) - 1)/3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)

a(n) = sumdiv(n, d, if (issquare(d), (-1)^((d%2)+1)*d)); \\ Michel Marcus, Aug 22 2021

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1]==2, 2 - (2^(2*floor(f[i,2]/2) + 2) - 1)/3, (f[i,1]^(2*floor(f[i,2]/2) + 2) - 1)/(f[i,1]^2 - 1)));} \\ Amiram Eldar, Nov 15 2022

Multiplicative with a(2^e) = 2 - (2^(2*floor(e/2) + 2) - 1)/3, and a(p^e) = (p^(2*floor(e/2) + 2) - 1)/(p^2 - 1) for p > 2. - Amiram Eldar, Nov 15 2022

cp's OEIS Frontend

A321552 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^7.

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A321557 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^12.

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

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A363022 Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^3.

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A321553 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^8.

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A321554 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^9.

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A321555 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^10.

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A321556 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^11.

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