A295883 Number of exponents that are 3 in the prime factorization of n.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Antti Karttunen, Nov 29 2017

Keywords

• nonn

Examples

			For n = 8 = 2^3, there is one exponent that is exactly 3, thus a(8) = 1.
For n = 216 = 2^3 * 3^3 there are two exponents that are exactly 3, thus a(216) = 2.
For n = 432 = 2^4 * 3^3, there is one exponent that is exactly 3, thus a(432) = 1.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A085541, A085964, A295659, A295662, A295663, A295884.

Programs

Mathematica

Array[Total@ Map[Boole[# == 3] &, FactorInteger[#][[All, -1]]] &, 120] (* Michael De Vlieger, Nov 29 2017 *)
Count[FactorInteger[#][[All,2]],3]&/@Range[120] (* Harvey P. Dale, Apr 13 2019 *)

PARI

a(n) = vecsum(apply(x->(x==3), factor(n)[,2])); \\ Michel Marcus, Jul 25 2022

Formula

Additive with a(p^3) = 1, a(p^e) = 0 when e <> 3.

a(n) = A295659(n) - A295884(n).

a(n) <= A295662(n) <= A295663(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (1/p^3 - 1/p^4) = A085541 - A085964 = 0.0977694995... . - Amiram Eldar, Jul 25 2022