A319089 a(n) = tau(n)^3, where tau is A000005.
1, 8, 8, 27, 8, 64, 8, 64, 27, 64, 8, 216, 8, 64, 64, 125, 8, 216, 8, 216, 64, 64, 8, 512, 27, 64, 64, 216, 8, 512, 8, 216, 64, 64, 64, 729, 8, 64, 64, 512, 8, 512, 8, 216, 216, 64, 8, 1000, 27, 216, 64, 216, 8, 512, 64, 512, 64, 64, 8, 1728, 8, 64, 216, 343
Offset: 1
Links
- Eric Weisstein's World of Mathematics, Divisor Function.
Programs
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Maple
with(numtheory): seq(tau(n)^3, n=1..100); # Ridouane Oudra, Mar 07 2023
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Mathematica
DivisorSigma[0, Range[100]]^3
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PARI
a(n) = numdiv(n)^3; \\ Altug Alkan, Sep 10 2018
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PARI
for(n=1, 100, print1(direuler(p=2, n, (1 + 4*X + X^2)/(1 - X)^4)[n], ", ")) \\ Vaclav Kotesovec, Mar 09 2023
Formula
Multiplicative with a(p^e) = (e+1)^3. - Amiram Eldar, Dec 31 2022
a(n) = Sum_{d1|n} Sum_{d2|n} tau(d1*d2). - Ridouane Oudra, Mar 07 2023
From Vaclav Kotesovec, Mar 09 2023: (Start)
Dirichlet g.f.: Product_{p prime} p^(2*s) * (1 + 4*p^s + p^(2*s)) / (p^s - 1)^4.
Dirichlet g.f.: zeta(s)^8 * Product_{p prime} (1 - 9/p^(2*s) + 16/p^(3*s) - 9/p^(4*s) + 1/p^(6*s)), (with a product that converges for s=1). (End)