login
A369935
The maximal exponent in the prime factorization of the numbers whose all exponents are squares (A197680).
4
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
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
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Feb 06 2024
STATUS
approved