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

1,12

Comments

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

Links

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

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.

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]

Prog

(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), ", "))); }

Crossrefs

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

Sequence in context: A110916 A185058 A368474 * A374325 A368169 A367418

Adjacent sequences: A369932 A369933 A369934 * A369936 A369937 A369938

Keyword

nonn,easy

Author

Amiram Eldar, Feb 06 2024

Status

approved