A374325 The maximal exponent in the prime factorization of the numbers whose maximal exponent in their prime factorization is a square.

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Jul 04 2024

Keywords

• nonn
• easy

Comments

First differs from A369935 at n = 95.

The first occurrence of k^2, for k = 0, 1, ..., is at 1, 2, 12, 335, 42563, ..., which is the position of 2^(k^2) at A369937.

Links

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

Crossrefs

Cf. A051903, A369935, A369937.

Similar sequences: A374324, A374326, A374327, A374328.

Programs

Mathematica

f[n_] := Module[{e = If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]}, If[IntegerQ@ Sqrt[e], e, Nothing]]; Array[f, 200]

PARI

lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = vecmax(factor(k)[, 2]); if(issquare(e), print1(e, ", ")));}

Formula

a(n) = A374326(n)^2.

a(n) = A051903(A369937(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k^2 * d(k) / Sum_{k>=1} d(k) = 1.19949058780036416105..., where d(k) = 1/zeta(k^2+1) - 1/zeta(k^2) for k>=2, and d(1) = 1/zeta(2).