A282254 - OEIS

%I #46 Nov 21 2024 07:06:25

%S 0,1,1026,59052,1050628,9765630,60587352,282475256,1075843080,

%T 3486961557,10019536380,25937424612,62041684656,137858491862,

%U 289819612656,576679982760,1101663313936,2015993900466,3577622557482,6131066257820,10260044315640

%N 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.

%C Multiplicative because A013957 is. - _Andrew Howroyd_, Jul 25 2018

%C 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

%H Seiichi Manyama, <a href="/A282254/b282254.txt">Table of n, a(n) for n = 0..1000</a>

%H Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Ramanujan_tau_function#Congruences_for_the_tau_function">Congruences for the tau function</a>.

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

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

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

%F 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

%F Multiplicative with a(p^e) = p^e*(p^(9*(e+1))-1)/(p^9-1). - _Jianing Song_, Aug 12 2020

%F From _Amiram Eldar_, Oct 30 2023: (Start)

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

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

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

%t Table[If[n>0, n * DivisorSigma[9, n], 0], {n, 0, 20}] (* _Indranil Ghosh_, Mar 12 2017 *)

%o (PARI) for(n=0, 20, print1(if(n==0, 0, n * sigma(n, 9)),", ")) \\ _Indranil Ghosh_, Mar 12 2017

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

%Y 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).

%Y Cf. A013957, A013668, A196292.

%K nonn,easy,mult

%O 0,3

%A _Seiichi Manyama_, Feb 10 2017