A376365 The number of distinct prime factors of the cubefree numbers.

0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 3, 1, 2, 2, 2, 2, 1, 2, 2, 1, 3, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 3, 1, 2, 2, 3, 1, 1, 2, 2, 2, 2, 3, 1, 2, 1, 3, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 3, 2, 1, 1, 3, 2, 1, 3, 2, 2, 2, 2, 2, 1, 2, 2, 2

Offset

1,6

Links

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

Sourabhashis Das, Wentang Kuo, and Yu-Ru Liu, Distribution of omega(n) over h-free and h-full numbers, arXiv:2409.10430 [math.NT], 2024. See Theorem 1.1.

Formula

a(n) = A001221(A004709(n)).

Sum_{A004709(k) <= x} a(k) = (6/Pi^2) * x * (log(log(x)) + B - C) + O(x/log(x)), where B is Mertens's constant (A077761) and C = Sum_{p prime} (p-1)/(p*(p^3-1)) = 0.10770743252352371604... (Das et al., 2024).

Mathematica

f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # < 3 &], Length[e], 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(#e, ", "))); }

Crossrefs

Cf. A001221, A004709, A059956, A072047, A077761, A376361, A376363, A376366.

Sequence in context: A103684 A105103 A377648 * A086669 A357906 A053574

Adjacent sequences: A376362 A376363 A376364 * A376366 A376367 A376368

Keyword

nonn,easy

Author

Amiram Eldar, Sep 21 2024

Status

approved