A281372 Coefficients in q-expansion of (E_2*E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.
0, 1, 18, 84, 292, 630, 1512, 2408, 4680, 6813, 11340, 14652, 24528, 28574, 43344, 52920, 74896, 83538, 122634, 130340, 183960, 202272, 263736, 279864, 393120, 393775, 514332, 551880, 703136, 707310, 952560, 923552, 1198368, 1230768, 1503684, 1517040, 1989396, 1874198, 2346120, 2400216, 2948400
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
Programs
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Magma
[0] cat [n*DivisorSigma(3, n): n in [1..50]]; // Vincenzo Librandi, Mar 01 2018
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Maple
with(gfun): with(numtheory); M:=100; E := proc(k) local n, t1; global M; t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..M+1); series(t1, q, M+1); end; e2:=E(2); e4:=E(4); e6:=E(6); t1:=series((e2*e4-e6)/720,q,M+1); seriestolist(t1); # alternative program seq(add(sigma[4](d)*phi(n/d), d in divisors(n)), n = 1..100); # Peter Bala, Jan 20 2024
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Mathematica
Table[If[n==0, 0, n * DivisorSigma[3, n]], {n, 0, 40}] (* Indranil Ghosh, Mar 11 2017 *) terms = 41; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[(Ei[2] Ei[4] - Ei[6])/720 + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *) -
PARI
for(n=0, 40, print1(if(n==0, 0, n * sigma(n, 3)), ", ")) \\ Indranil Ghosh, Mar 11 2017
Formula
a(n) = A145094(n)/240 for n > 0. - Seiichi Manyama, Feb 04 2017
G.f.: phi_{4, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}. - Seiichi Manyama, Feb 04 2017
a(n) = n*A001158(n) for n > 0. - Seiichi Manyama, Feb 18 2017
G.f.: x*f'(x), where f(x) = Sum_{k>=1} k^3*x^k/(1 - x^k). - Ilya Gutkovskiy, Aug 31 2017
Sum_{k=1..n} a(k) ~ Pi^4 * n^5 / 450. - Vaclav Kotesovec, May 09 2022
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(3*e+3)-1)/(p^3-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-4). (End)
a(n) = Sum_{k = 1..n} sigma_4( gcd(k, n) ) = Sum_{d divides n} sigma_4(d) * phi(n/d). - Peter Bala, Jan 19 2024
a(n) = Sum_{1 <= i, j, k, l <= n} sigma_1( gcd(i, j, k, l, n) ) = Sum_{d divides n} sigma_1(d) * J_4(n/d), where the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 22 2024
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