A376362 The number of unitary divisors that are squares of primes applied to the powerful numbers.

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

Offset

1,9

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. See Theorem 1.3.

Index entries for sequences related to powerful numbers.

Formula

a(n) = A369427(A001694(n)).

Sum_{A001694(k) <= x} a(k) = c * sqrt(x) * (log(log(x)) + B - log(2) - L(2, 4)) + O(sqrt(x)/log(x)), where c = zeta(3/2)/zeta(3) (A090699), B is Mertens's constant (A077761), L(h, r) = Sum_{p prime} 1/(p^(r/h - 1) * (p - p^(1 - 1/h) + 1)), and L(2, 4) = 0.57937575954505652569... (Das et al., 2024).

Mathematica

f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # > 1 &], Count[e, 2], Nothing]]; Array[f, 3500]

Prog

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

Crossrefs

Cf. A001694, A077761, A090699, A369427, A376361, A376364, A376366.

Sequence in context: A068906 A162514 A364039 * A166347 A055300 A335392

Adjacent sequences: A376359 A376360 A376361 * A376363 A376364 A376365

Keyword

nonn,easy

Author

Amiram Eldar, Sep 21 2024

Status

approved