A376366 The number of non-unitary prime divisors of the cubefree numbers.

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1

Offset

1,31

Links

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

Sourabhashis Das, Wentang Kuo, and Yu-Ru Liu, On the number of prime factors with a given multiplicity over h-free and h-full numbers, arXiv:2409.11275 [math.NT], 2024. Theorem 1.2.

Formula

a(n) = A056170(A004709(n)).

a(n) = A369427(A004709(n)).

Sum_{A004709(k) <= x} a(k) = c * x + O(sqrt(x)/log(x)), where c = (1/zeta(3)) * Sum_{p prime} (p*(p-1)/(p^3-1)) = 0.24833233043359932037... (Das et al., 2024).

Mathematica

f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # < 3 &], Count[e, 2], Nothing]]; Array[f, 150]

Prog

(PARI) lista(kmax) = {my(e, is); for(k = 1, kmax, e = factor(k)[, 2]; is = 1; for(i = 1, #e, if(e[i] > 2, is = 0; break)); if(is, print1(#select(x -> x == 2, e), ", "))); }

Crossrefs

Cf. A002117, A004709, A056170, A088453, A369427, A376362, A376365.

Sequence in context: A107329 A263717 A230279 * A085859 A218218 A332006

Adjacent sequences: A376363 A376364 A376365 * A376367 A376368 A376369

Keyword

nonn,easy

Author

Amiram Eldar, Sep 21 2024

Status

approved