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A282254
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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.
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7
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0, 1, 1026, 59052, 1050628, 9765630, 60587352, 282475256, 1075843080, 3486961557, 10019536380, 25937424612, 62041684656, 137858491862, 289819612656, 576679982760, 1101663313936, 2015993900466, 3577622557482, 6131066257820, 10260044315640
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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G.f.: phi_{10, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
If p is a prime, a(p) = p^10 + p = A196292(p).
Multiplicative with a(p^e) = p^e*(p^(9*(e+1))-1)/(p^9-1). - Jianing Song, Aug 12 2020
Dirichlet g.f.: zeta(s-1)*zeta(s-10).
Sum_{k=1..n} a(k) ~ zeta(10) * n^11 / 11. (End)
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EXAMPLE
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a(6) = 1^10*6^1 + 2^10*3^1 + 3^10*2^1 + 6^10*1^1 = 60587352.
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MATHEMATICA
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Table[If[n>0, n * DivisorSigma[9, n], 0], {n, 0, 20}] (* Indranil Ghosh, Mar 12 2017 *)
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PROG
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(PARI) for(n=0, 20, print1(if(n==0, 0, n * sigma(n, 9)), ", ")) \\ Indranil Ghosh, Mar 12 2017
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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