A321543 -id:A321543 - cp's OEIS frontend

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

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)

A321828 a(n) = Sum_{d|n, d==1 mod 4} d^12 - Sum_{d|n, d==3 mod 4} d^12.

Original entry on oeis.org

1, 1, -531440, 1, 244140626, -531440, -13841287200, 1, 282429005041, 244140626, -3138428376720, -531440, 23298085122482, -13841287200, -129746094281440, 1, 582622237229762, 282429005041, -2213314919066160, 244140626, 7355813669568000

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).
Index entries for sequences mentioned by Glaisher.

Column k=12 of A322143.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A000265.

s[n_, r_] := DivisorSum[n, #^12 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^12)^(e+1)-1)/(p^12-1), ((-p^12)^(e+1)-1)/(-p^12-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)

apply( A321828(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^12), [1..40]) \\ M. F. Hasler, Nov 26 2018

a(n) = a(A000265(n)). - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^12*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018
Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^12)^(e+1)-1)/(p^12-1) if p == 1 (mod 4) and ((-p^12)^(e+1)-1)/(-p^12-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023
a(n) = Sum_{d|n} d^12*sin(d*Pi/2). - Ridouane Oudra, Sep 08 2024

A363630 Expansion of Sum_{k>0} (1/(1+x^k)^3 - 1).

Original entry on oeis.org

-3, 3, -13, 18, -24, 21, -39, 63, -68, 48, -81, 127, -108, 87, -170, 216, -174, 156, -213, 294, -302, 201, -303, 497, -375, 276, -474, 537, -468, 426, -531, 777, -686, 462, -726, 965, -744, 573, -938, 1200, -906, 798, -993, 1251, -1306, 831, -1179, 1875, -1314, 1023, -1562, 1722, -1488, 1290, -1698

Offset: 1

Seiichi Manyama, Jun 12 2023

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

Cf. A002129, A048272, A321543, A363629, A363631.

Cf. A320900, A363022, A363615.

a[n_] := DivisorSum[n, (-1)^#*Binomial[# + 2, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 18 2023 *)

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

G.f.: Sum_{k>0} binomial(k+2,2) * (-x)^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^d * binomial(d+2,2).
a(n) = -(A321543(n) + 3*A002129(n) + 2*A048272(n)) / 2. - Amiram Eldar, Jan 04 2025

A321544 a(n) = Sum_{d|n} (-1)^(d-1)*d^5.

Original entry on oeis.org

1, -31, 244, -1055, 3126, -7564, 16808, -33823, 59293, -96906, 161052, -257420, 371294, -521048, 762744, -1082399, 1419858, -1838083, 2476100, -3297930, 4101152, -4992612, 6436344, -8252812, 9768751, -11510114, 14408200, -17732440, 20511150, -23645064, 28629152, -34636831, 39296688, -44015598

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.

Divisor sums Sum_{d|n} (-1)^(d-1)*d^k: A048272 (k = 0), A002129 (k = 1), A321543 (k = 2), A138503 (k = 3), A279395 (k = 4, unsigned), A321545 - A321551 (k = 6 to k = 12).

Cf. A321552 - A321565, A321807 - A321836 for similar sequences.

with(numtheory):
a := n -> add( (-1)^(d-1)*d^5, d in divisors(n) ): seq(a(n), n = 1..40);
#  Peter Bala, Jan 11 2021

f[p_, e_] := (p^(5*e + 5) - 1)/(p^5 - 1); f[2, e_] := 2 - (2^(5*e + 5) - 1)/31; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 35] (* Amiram Eldar, Nov 04 2022 *)

apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^5), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} (-1)^(k-1)*k^5*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 23 2018
G.f.: Sum_{n >= 1} x^n*(x^(4*n) - 26*x^(3*n) + 66*x^(2*n) - 26*x^n + 1)/(1 + x^n)^6 (note [1,26,66,26,1] is row 5 of A008292). - Peter Bala, Jan 11 2021
Multiplicative with a(2^e) = 2 - (2^(5*e + 5) - 1)/31, and a(p^e) = (p^(5*e + 5) - 1)/(p^5 - 1) for p > 2. - Amiram Eldar, Nov 04 2022

A321545 a(n) = Sum_{d|n} (-1)^(d-1)*d^6.

Original entry on oeis.org

1, -63, 730, -4159, 15626, -45990, 117650, -266303, 532171, -984438, 1771562, -3036070, 4826810, -7411950, 11406980, -17043519, 24137570, -33526773, 47045882, -64988534, 85884500, -111608406, 148035890, -194401190, 244156251, -304089030, 387952660, -489306350, 594823322, -718639740, 887503682

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.

f[p_, e_] := (p^(6*e + 6) - 1)/(p^6 - 1); f[2, e_] := 2 - (2^(6*e + 6) - 1)/63; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 04 2022 *)

apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^6), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} (-1)^(k-1)*k^6*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 23 2018

Multiplicative with a(2^e) = 2 - (2^(6*e + 6) - 1)/63, and a(p^e) = (p^(6*e + 6) - 1)/(p^6 - 1) for p > 2. - Amiram Eldar, Nov 04 2022

A321546 a(n) = Sum_{d|n} (-1)^(d-1)*d^7.

Original entry on oeis.org

1, -127, 2188, -16511, 78126, -277876, 823544, -2113663, 4785157, -9922002, 19487172, -36126068, 62748518, -104590088, 170939688, -270549119, 410338674, -607714939, 893871740, -1289938386, 1801914272, -2474870844, 3404825448, -4624694644, 6103593751, -7969061786, 10465138360, -13597534984, 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.

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

f[p_, e_] := (p^(7*e + 7) - 1)/(p^7 - 1); f[2, e_] := 2 - (2^(7*e + 7) - 1)/127; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 04 2022 *)

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

G.f.: Sum_{k>=1} (-1)^(k-1)*k^7*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 23 2018

Multiplicative with a(2^e) = 2 - (2^(7*e + 7) - 1)/127, and a(p^e) = (p^(7*e + 7) - 1)/(p^7 - 1) for p > 2. - Amiram Eldar, Nov 04 2022

A321559 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^3.

Original entry on oeis.org

1, -9, 28, -57, 126, -252, 344, -441, 757, -1134, 1332, -1596, 2198, -3096, 3528, -3513, 4914, -6813, 6860, -7182, 9632, -11988, 12168, -12348, 15751, -19782, 20440, -19608, 24390, -31752, 29792, -28089, 37296, -44226, 43344, -43149, 50654

Offset: 1

N. J. A. Sloane, Nov 23 2018

G. C. Greubel, Table of n, a(n) for n = 1..1000
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=3 of A322083.

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

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(-1)^(k+1)*k^3*x^k/(1 + x^k) : k in [1..2*m]]) )); // G. C. Greubel, Nov 28 2018

a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^3 &]; Array[a, 50] (* Amiram Eldar, Nov 27 2018 *)

apply( A321559(n)=sumdiv(n, d, (-1)^(n\d-d)*d^3), [1..30]) \\ M. F. Hasler, Nov 26 2018

s=(sum((-1)^(k+1)*k^3*x^k/(1 + x^k)  for k in (1..50))).series(x, 50); a = s.coefficients(x, sparse=False); a[1:] # G. C. Greubel, Nov 28 2018

G.f.: Sum_{k>=1} (-1)^(k+1)*k^3*x^k/(1 + x^k). - Ilya Gutkovskiy, Nov 27 2018
From Peter Bala, Jan 29 2022: (Start)
Multiplicative with a(2^k) = - 3*(2^(3*k+1) + 5)/7 for k >= 1 and a(p^k) = (p^(3*k+3) - 1)/(p^3 - 1) for odd prime p.
n^3 = (-1)^(n+1)*Sum_{d divides n} A067856(n/d)*a(d). (End)

A321560 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^4.

Original entry on oeis.org

1, -17, 82, -241, 626, -1394, 2402, -3825, 6643, -10642, 14642, -19762, 28562, -40834, 51332, -61169, 83522, -112931, 130322, -150866, 196964, -248914, 279842, -313650, 391251, -485554, 538084, -578882, 707282, -872644, 923522, -978673, 1200644

Offset: 1

N. J. A. Sloane, Nov 23 2018

G. C. Greubel, Table of n, a(n) for n = 1..1000
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=4 of A322083.

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

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(-1)^(k+1)*k^4*x^k/(1 + x^k) : k in [1..2*m]]) )); // G. C. Greubel, Nov 28 2018

a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^4 &]; Array[a, 50] (* Amiram Eldar, Nov 27 2018 *)

apply( A321560(n)=sumdiv(n, d, (-1)^(n\d-d)*d^4), [1..30]) \\ M. F. Hasler, Nov 26 2018

s=(sum((-1)^(k+1)*k^4*x^k/(1 + x^k)  for k in (1..50))).series(x, 50); a = s.coefficients(x, sparse=False); a[1:] # G. C. Greubel, Nov 28 2018

G.f.: Sum_{k>=1} (-1)^(k+1)*k^4*x^k/(1 + x^k). - Ilya Gutkovskiy, Nov 27 2018

Multiplicative with a(2^e) = -(7*2^(4*e+1) + 31)/15, and a(p^e) = (p^(4*e+4) - 1)/(p^4 - 1) for p > 2. - Amiram Eldar, Nov 22 2022

A321561 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^5.

Original entry on oeis.org

1, -33, 244, -993, 3126, -8052, 16808, -31713, 59293, -103158, 161052, -242292, 371294, -554664, 762744, -1014753, 1419858, -1956669, 2476100, -3104118, 4101152, -5314716, 6436344, -7737972, 9768751, -12252702, 14408200, -16690344, 20511150

Offset: 1

N. J. A. Sloane, Nov 23 2018

G. C. Greubel, Table of n, a(n) for n = 1..1000
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=5 of A322083.

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

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(-1)^(k+1)*k^5*x^k/(1 + x^k) : k in [1..2*m]]) )); // G. C. Greubel, Nov 28 2018

a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^5 &]; Array[a, 50] (* Amiram Eldar, Nov 27 2018 *)

apply( A321561(n)=sumdiv(n, d, (-1)^(n\d-d)*d^5), [1..30]) \\ M. F. Hasler, Nov 26 2018

s=(sum((-1)^(k+1)*k^5*x^k/(1 + x^k)  for k in (1..50))).series(x, 50); a = s.coefficients(x, sparse=False); a[1:] # G. C. Greubel, Nov 28 2018

G.f.: Sum_{k>=1} (-1)^(k+1)*k^5*x^k/(1 + x^k). - Ilya Gutkovskiy, Nov 27 2018

Multiplicative with a(2^e) = -3*(5*2^(5*e+1) + 21)/31, and a(p^e) = (p^(5*e+5) - 1)/(p^5 - 1) for p > 2. - Amiram Eldar, Nov 22 2022

A321562 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^6.

Original entry on oeis.org

1, -65, 730, -4033, 15626, -47450, 117650, -257985, 532171, -1015690, 1771562, -2944090, 4826810, -7647250, 11406980, -16510913, 24137570, -34591115, 47045882, -63019658, 85884500, -115151530, 148035890, -188329050, 244156251, -313742650

Offset: 1

N. J. A. Sloane, Nov 23 2018

G. C. Greubel, Table of n, a(n) for n = 1..1000
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=6 of A322083.

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

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(-1)^(k+1)*k^6*x^k/(1 + x^k) : k in [1..2*m]]) )); // G. C. Greubel, Nov 28 2018

a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^6 &]; Array[a, 50] (* Amiram Eldar, Nov 27 2018 *)

apply( A321562(n)=sumdiv(n, d, (-1)^(n\d-d)*d^6), [1..30]) \\ M. F. Hasler, Nov 26 2018

s=(sum((-1)^(k+1)*k^6*x^k/(1 + x^k)  for k in (1..50))).series(x, 50); a = s.coefficients(x, sparse=False); a[1:] # G. C. Greubel, Nov 28 2018

G.f.: Sum_{k>=1} (-1)^(k+1)*k^6*x^k/(1 + x^k). - Ilya Gutkovskiy, Nov 27 2018

Multiplicative with a(2^e) = -(31*2^(6*e+1) + 127)/63, and a(p^e) = (p^(6*e+6) - 1)/(p^6 - 1) for p > 2. - Amiram Eldar, Nov 22 2022

cp's OEIS Frontend

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

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A321828 a(n) = Sum_{d|n, d==1 mod 4} d^12 - Sum_{d|n, d==3 mod 4} d^12.

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A363630 Expansion of Sum_{k>0} (1/(1+x^k)^3 - 1).

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A321544 a(n) = Sum_{d|n} (-1)^(d-1)*d^5.

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A321545 a(n) = Sum_{d|n} (-1)^(d-1)*d^6.

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A321546 a(n) = Sum_{d|n} (-1)^(d-1)*d^7.

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A321559 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^3.

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A321560 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^4.

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