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

Author

Seiichi Manyama, Feb 10 2017

Keywords

• nonn
• easy
• mult

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

Examples

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• Wikipedia, Congruences for the tau function.

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.

Programs

Mathematica

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

PARI

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

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)