A282254 Coefficients in q-expansion of (3*E_4^3 + 2*E_6^2 - 5*E_2*E_4*E_6)/1584, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

0, 1, 1026, 59052, 1050628, 9765630, 60587352, 282475256, 1075843080, 3486961557, 10019536380, 25937424612, 62041684656, 137858491862, 289819612656, 576679982760, 1101663313936, 2015993900466, 3577622557482, 6131066257820, 10260044315640

Offset

0,3

Comments

Multiplicative because A013957 is. - Andrew Howroyd, Jul 25 2018

D. H. Lehmer shows that a(n) == tau(n) (mod 7) for n > 0, where tau is Ramanujan's tau function (A000594). Furthermore, if n == 3, 5, 6 (mod 7) then a(n) == tau(n) (mod 49). See the Wikipedia link below. - Jianing Song, Aug 12 2020

Links

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

Wikipedia, Congruences for the tau function.

Formula

G.f.: phi_{10, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

a(n) = (3*A008411(n) + 2*A280869(n) - 5*A282102(n))/1584.

If p is a prime, a(p) = p^10 + p = A196292(p).

a(n) = n*A013957(n) for n > 0, where A013957(n) is sigma_9(n), the sum of the 9th powers of the divisors of n. - Seiichi Manyama, Feb 18 2017

Multiplicative with a(p^e) = p^e*(p^(9*(e+1))-1)/(p^9-1). - Jianing Song, Aug 12 2020

From Amiram Eldar, Oct 30 2023: (Start)

Dirichlet g.f.: zeta(s-1)*zeta(s-10).

Sum_{k=1..n} a(k) ~ zeta(10) * n^11 / 11. (End)

Example

a(6) = 1^10*6^1 + 2^10*3^1 + 3^10*2^1 + 6^10*1^1 = 60587352.

Mathematica

Table[If[n>0, n * DivisorSigma[9, n], 0], {n, 0, 20}] (* Indranil Ghosh, Mar 12 2017 *)

Prog

(PARI) for(n=0, 20, print1(if(n==0, 0, n * sigma(n, 9)), ", ")) \\ Indranil Ghosh, Mar 12 2017

Crossrefs

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}, A282060 (phi_{8, 1}), this sequence (phi_{10, 1}).

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008411 (E_4^3), A280869 (E_6^2), A282102 (E_2*E_4*E_6).

Cf. A013957, A013668, A196292.

Sequence in context: A004607 A361811 A221008 * A229332 A253710 A271761

Adjacent sequences: A282251 A282252 A282253 * A282255 A282256 A282257

Keyword

nonn,easy,mult

Author

Seiichi Manyama, Feb 10 2017

Status

approved