A282596 -id:A282596 - cp's OEIS frontend

A386787 a(n) = n^4*sigma_7(n).

0, 1, 2064, 177228, 4227328, 48828750, 365798592, 1977329144, 8657571840, 31395415077, 100782540000, 285311685252, 749200886784, 1792160422598, 4081207353216, 8653821705000, 17730707193856, 34271896391154, 64800136718928, 116490259028540, 206415142080000, 350438089532832

Offset: 0

Vaclav Kotesovec, Aug 02 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..7500

Cf. A280022, A386783, A280025, A386784, A386785, A386786, A386788.
Cf. A013955, A282060, A282753, A386781.
Cf. A386815, A386816, A282012, A386817, A282596, A386818, A282287.

[0] cat [n^4*DivisorSigma(7, n): n in [1..35]]; // Vincenzo Librandi, Aug 03 2025

Table[n^4*DivisorSigma[7, n], {n, 0, 30}]
(* or *)
nmax = 30; CoefficientList[Series[Sum[k^4*x^k*(1 + 2036*x^k + 152637*x^(2*k) + 2203488*x^(3*k) + 9738114*x^(4*k) + 15724248*x^(5*k) + 9738114*x^(6*k) + 2203488*x^(7*k) + 152637*x^(8*k) + 2036*x^(9*k) + x^(10*k))/(1 - x^k)^12, {k, 1, nmax}], {x, 0, nmax}], x]
(* or *)
terms = 30; E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}]; E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}]; E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}]; CoefficientList[Series[(33*E2[x]^4*E4[x]^2 + 110*E2[x]^2*E4[x]^3 + 13*E4[x]^4 - 132*E2[x]^3*E4[x]*E6[x] - 132*E2[x]*E4[x]^2*E6[x] + 88*E2[x]^2*E6[x]^2 + 20*E4[x]*E6[x]^2)/41472, {x, 0, terms}], x]

G.f.: Sum_{k>=1} k^4*x^k*(1 + 2036*x^k + 152637*x^(2*k) + 2203488*x^(3*k) + 9738114*x^(4*k) + 15724248*x^(5*k) + 9738114*x^(6*k) + 2203488*x^(7*k) + 152637*x^(8*k) + 2036*x^(9*k) + x^(10*k))/(1 - x^k)^12.
a(n) = (33*A386815(n) + 110*A386816(n) + 13*A282012(n) - 132*A386817(n) - 132*A282596(n) + 88*A386818(n) + 20*A282287(n))/41472.
a(n) = n^4*A013955(n).
Dirichlet g.f.: zeta(s-4)*zeta(s-11). - R. J. Mathar, Aug 03 2025

A282792 Coefficients in q-expansion of E_2^2E_4E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

1, -312, -122328, 1193376, 120735336, 123318576, -26119268064, -383848045248, -3132125965080, -18290795499096, -84925855577232, -331983655889184, -1133781877844448, -3470165144567184, -9697162366507968, -25093220330304576, -60786860467926552

Offset: 0

Seiichi Manyama, Feb 21 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A282102 (E_2*E_4*E_6), this sequence (E_2^2*E_4*E_6), A282596 (E_2*E_4^2*E_6), A282547 (E_2*E_4*E_6^2).

terms = 17;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^2*E6[x]*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282597 Expansion of phi_{14, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 16386, 4782972, 268468228, 6103515630, 78373779192, 678223072856, 4398583447560, 22876806803877, 100012207113180, 379749833583252, 1284076017413616, 3937376385699302, 11113363271818416, 29192944359852360, 72066391204823056, 168377826559400946

Offset: 0

Seiichi Manyama, Feb 19 2017

Multiplicative because A013961 is. - Andrew Howroyd, Jul 25 2018

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

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), A282060 (phi_{8, 1}), A282254 (phi_{10, 1}), A282548 (phi_{12, 1}), this sequence (phi_{14, 1}).
Cf. A282012 (E_4^4), A282287 (E_4*E_6^2), A282596 (E_2*E_4^2*E_6).
Cf. A013672.

Table[n * DivisorSigma[13, n], {n, 0, 17}] (* Amiram Eldar, Sep 06 2023 *)

a(n) = if(n < 1, 0, n*sigma(n, 13)) \\ Andrew Howroyd, Jul 25 2018

a(n) = n*A013961(n) for n > 0.
a(n) = (3*A282012(n) + 4*A282287(n) - 7*A282596(n))/144.
Sum_{k=1..n} a(k) ~ zeta(14) * n^15 / 15. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(13*e+13)-1)/(p^13-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-14). (End)

A319134 Expansion of -((25E_4^4 - 49E_6^2E4) + 48E_6E_4^2E_2 + (-49E_4^3 + 25E_6^2)E_2^2)/(3657830400delta^2) where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively and delta is A000594.

Original entry on oeis.org

1, 86, 3750, 109672, 2419462, 43021728, 643548464, 8343640624, 95835049605, 991606081332, 9364586280842, 81571540591968, 661034448807902, 5019357866562208, 35927279225314344, 243657157464337888, 1572638456431119570, 9696997279843999470, 57313953586222481126, 325672739267123628976

Offset: 1

Seiichi Manyama, Sep 11 2018

nonn

			((25*E_4^4 - 49*E_6^2*E4) + 48*E_6*E_4^2*E_2 + (-49*E_4^3 + 25*E_6^2)*E_2^2)/(delta^2) =  - 3657830400*q - 314573414400*q^2 - 13716864000000*q^3 - 401161575628800*q^4 - ... .

Seiichi Manyama, Table of n, a(n) for n = 1..5000
H. Cohn, A. Kumar, S. Miller, D. Radchenko, M. Viazovska, The sphere packing problem in dimension 24, Annals of Mathematics, 185 (3) (2017), 1017-1033.
Wikipedia, Sphere packing

Cf. A000594, A006352 (E_2), A004009 (E_4), A013973 (E_6), A082558, A281373,

About the numerator: A282012 (E_4^4), A282287 (E_6^2*E_4), A282596 (E_6*E_4^2*E_2), A008411 (E_4^3), A280869 (E_6^2), A281374 (E_2^2).

nmax = 25; E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, nmax + 1}] + O[x]^(nmax + 1); E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, nmax + 1}] + O[x]^(nmax + 1); E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, nmax + 1}] + O[x]^(nmax + 1); Rest[CoefficientList[Series[-((25*E4[x]^4 - 49*E6[x]^2*E4[x]) + 48*E6[x]*E4[x]^2*E2[x] + (-49*E4[x]^3 + 25*E6[x]^2)* E2[x]^2) / (3657830400 * x^2 * QPochhammer[x]^48), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 12 2018 *)

a(n) ~ exp(4*Pi*sqrt(2*n)) / (132300 * 2^(1/4) * Pi^2 * n^(23/4)). - Vaclav Kotesovec, Sep 12 2018

cp's OEIS Frontend

A386787 a(n) = n^4*sigma_7(n).

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A282792 Coefficients in q-expansion of E_2^2E_4E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A282597 Expansion of phi_{14, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

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A319134 Expansion of -((25E_4^4 - 49E_6^2E4) + 48E_6E_4^2E_2 + (-49E_4^3 + 25E_6^2)E_2^2)/(3657830400delta^2) where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively and delta is A000594.

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cp's OEIS Frontend

A386787 a(n) = n^4*sigma_7(n).

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Author

Keywords

Links

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Programs

Magma

Mathematica

Formula

A282792 Coefficients in q-expansion of E_2^2*E_4*E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A282597 Expansion of phi_{14, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

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A319134 Expansion of -((25*E_4^4 - 49*E_6^2*E4) + 48*E_6*E_4^2*E_2 + (-49*E_4^3 + 25*E_6^2)*E_2^2)/(3657830400*delta^2) where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively and delta is A000594.

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A282792 Coefficients in q-expansion of E_2^2E_4E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

A319134 Expansion of -((25E_4^4 - 49E_6^2E4) + 48E_6E_4^2E_2 + (-49E_4^3 + 25E_6^2)E_2^2)/(3657830400delta^2) where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively and delta is A000594.