A376363 The number of distinct prime factors of the cubefull numbers.

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

Offset

1,10

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 1.2.

Index entries for sequences related to powerful numbers.

Formula

a(n) = A001221(A036966(n)).

Sum_{A036966(k) <= x} a(k) = c * x^(1/3) * (log(log(x)) + B - log(3) + L(3, 4) - L(3, 6)) + O(x^(1/3)/log(x)), where c = A362974, B is Mertens's constant (A077761), L(h, r) = Sum_{p prime} 1/(p^(r/h - 1) * (p - p^(1 - 1/h) + 1)), L(3, 4) = 1.65235055631578303808..., and L(3, 6) = 0.67060646664392140547... (Das et al., 2024).

Mathematica

f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # > 2 &], Length[e], Nothing]]; Array[f, 60000]

Prog

(PARI) lista(kmax) = {my(e, is); for(k = 1, kmax, e = factor(k)[, 2]; is = 1; for(i = 1, #e, if(e[i] < 3, is = 0; break)); if(is, print1(#e, ", "))); }

Crossrefs

Cf. A001221, A036966, A072047, A077761, A362974, A376361, A376364, A376365.

Sequence in context: A012257 A220464 A215975 * A071891 A359306 A332761

Adjacent sequences: A376360 A376361 A376362 * A376364 A376365 A376366

Keyword

nonn,easy

Author

Amiram Eldar, Sep 21 2024

Status

approved