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A361147
a(n) = sigma(n)^3.
2
1, 27, 64, 343, 216, 1728, 512, 3375, 2197, 5832, 1728, 21952, 2744, 13824, 13824, 29791, 5832, 59319, 8000, 74088, 32768, 46656, 13824, 216000, 29791, 74088, 64000, 175616, 27000, 373248, 32768, 250047, 110592, 157464, 110592, 753571, 54872, 216000, 175616
OFFSET
1,2
FORMULA
Multiplicative with a(p^e) = ((p^(e+1)-1)/(p-1))^3.
Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-2) * zeta(s-3) * Product_{primes p} (1 + 1/p^(2*s-3) + 2/p^(s-1) + 2/p^(s-2)).
Sum_{k=1..n} a(k) ~ c * Pi^6 * zeta(3) * n^4 / 2160, where c = Product_{primes p} (1 + 2/p^2 + 2/p^3 + 1/p^5) = 2.83598357433419286770442457158038489640898183...
a(n) = A000578(A000203(n)).
MATHEMATICA
Table[DivisorSigma[1, n]^3, {n, 1, 50}]
PROG
(PARI) a(n) = sigma(n)^3;
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + p*X*(2 + 2*p + p^2*X)) / ((1-X)*(1-p*X)*(1-p^2*X)*(1-p^3*X)))[n], ", "))
KEYWORD
nonn,mult
AUTHOR
Vaclav Kotesovec, Mar 02 2023
STATUS
approved