A360970 Multiplicative with a(p^e) = e^3, p prime and e > 0.

1, 1, 1, 8, 1, 1, 1, 27, 8, 1, 1, 8, 1, 1, 1, 64, 1, 8, 1, 8, 1, 1, 1, 27, 8, 1, 27, 8, 1, 1, 1, 125, 1, 1, 1, 64, 1, 1, 1, 27, 1, 1, 1, 8, 8, 1, 1, 64, 8, 8, 1, 8, 1, 27, 1, 27, 1, 1, 1, 8, 1, 1, 8, 216, 1, 1, 1, 8, 1, 1, 1, 216, 1, 1, 8, 8, 1, 1, 1, 64, 64

Offset: 1

Author

Vaclav Kotesovec, Feb 27 2023

Keywords

• nonn
• mult

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Graph - the asymptotic ratio (10^9 terms)

Crossrefs

Cf. A005361, A226602, A360969, A361132.

Programs

Maple

f:= proc(n) local t;
  mul(t^3, t = ifactors(n)[2][..,2]);
end proc:
map(f, [$1..100]); # Robert Israel, Mar 29 2023

Mathematica

g[p_, e_] := e^3; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 - 3*X + 10*X^2 - 3*X^3 + X^4)/(1-X)^4)[n], ", "))

Formula

Dirichlet g.f.: zeta(s) * Product_{primes p} (1 + (7*p^(2*s) - 2*p^s + 1) / (p^s*(p^s - 1)^3)).

Sum_{k=1..n} a(k) ~ c * n, where c = Product_{primes p} (1 + (7*p^2 - 2*p + 1) / (p*(p-1)^3)) = 109.601930729008995813857898403091253809628920963774227252953...

a(n) = A005361(n)^3.