A382966 The number of non-unitary prime divisors of the n-th biquadratefree number that is not cubefree.

1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Author

Amiram Eldar, Apr 10 2025

Keywords

• nonn
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002117, A013662, A056170, A375072, A376366, A382425, A382965, A382968.

Programs

Mathematica

f[k_] := If[k == 1, Nothing, Module[{e = FactorInteger[k][[;; , 2]]}, If[Max[e] == 3, Count[e, _?(# > 1 &)], Nothing]]]; Array[f, 1000]

PARI

list(lim) = {my(e); for(k = 2, lim, e = factor(k)[, 2]; if(vecmax(e) == 3, print1(#select(x -> x > 1, e), ", ")));}

Formula

a(n) = A056170(A375072(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = ((1/zeta(4)) * Sum_{p prime} (1/(p^2+1)) - (1/zeta(3)) * Sum_{p prime} ((p-1)/(p^3-1))) / (1/zeta(4) - 1/zeta(3)) = 1.20757893653588072073... .