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

%I A360970 #19 Mar 30 2023 02:39:26
%S A360970 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,
%T A360970 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,
%U A360970 8,216,1,1,1,8,1,1,1,216,1,1,8,8,1,1,1,64,64
%N A360970 Multiplicative with a(p^e) = e^3, p prime and e > 0.
%H A360970 Robert Israel, <a href="/A360970/b360970.txt">Table of n, a(n) for n = 1..10000</a>
%H A360970 Vaclav Kotesovec, <a href="/A360970/a360970.jpg">Graph - the asymptotic ratio (10^9 terms)</a>
%F A360970 Dirichlet g.f.: zeta(s) * Product_{primes p} (1 + (7*p^(2*s) - 2*p^s + 1) / (p^s*(p^s - 1)^3)).
%F A360970 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...
%F A360970 a(n) = A005361(n)^3.
%p A360970 f:= proc(n) local t;
%p A360970   mul(t^3, t = ifactors(n)[2][..,2]);
%p A360970 end proc:
%p A360970 map(f, [$1..100]); # _Robert Israel_, Mar 29 2023
%t A360970 g[p_, e_] := e^3; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]
%o A360970 (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], ", "))
%Y A360970 Cf. A005361, A226602, A360969, A361132.
%K A360970 nonn,mult
%O A360970 1,4
%A A360970 _Vaclav Kotesovec_, Feb 27 2023