A368474 Product of exponents of prime factorization of the numbers whose exponents in their prime power factorization are squares (A197680).

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, 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

All the terms are squares (A000290).

The first position of k^2, for k = 1, 2, ..., is 1, 12, 331, 834, 21512290, 26588, ..., which is the position of A085629(k^2) in A197680.

Links

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

Formula

a(n) = A005361(A197680(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/d) * Product_{p prime} (1 + Sum_{k>=1} k^2/p^(k^2)) = 1.16776748073813763932..., where d = A357016 is the asymptotic density of A197680.

Mathematica

f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, IntegerQ[Sqrt[#]] &], Times @@ e, Nothing]]; Array[f, 150]

Prog

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

Crossrefs

Cf. A000290, A005361, A085629, A197680, A357016.

Similar sequences: A322327, A368472, A368473.

Sequence in context: A103524 A110916 A185058 * A369935 A374325 A368169

Adjacent sequences: A368471 A368472 A368473 * A368475 A368476 A368477

Keyword

nonn,easy

Author

Amiram Eldar, Dec 26 2023

Status

approved