A369935 The maximal exponent in the prime factorization of the numbers whose all exponents are squares (A197680).

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, Feb 06 2024

Keywords

• nonn
• easy

Comments

Differs from A368474 at n = 1, 834, 4154, 5822, 6417, ... .

Links

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

Crossrefs

Cf. A000290, A051903, A197680, A357016, A368474, A369936.

Programs

Mathematica

squareQ[n_] := IntegerQ[Sqrt[n]]; f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, squareQ], Max @@ e, Nothing]]; f[1] = 0; Array[f, 150]

PARI

lista(kmax) = {my(e, q); print1(0, ", "); for(k = 2, kmax, e = factor(k)[, 2]; q = 1; for(i = 1, #e, if(!issquare(e[i]), q = 0; break)); if(q, print1(vecmax(e), ", ")));}

Formula

a(n) = A051903(A197680(n)).

a(n) = A369936(n)^2.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (k^2 * (d(k) - d(k-1))) / A357016 = 1.16184898017948977008..., where d(k) = Product_{p prime} (1 - 1/p^2 + Sum_{i=2..k} (1/p^(i^2)-1/p^(i^2+1))) for k >= 1, and d(0) = 0.