A375341 The maximum exponent in the prime factorization of the numbers that have exactly one non-unitary prime factor.

2, 3, 2, 2, 4, 2, 2, 3, 2, 3, 2, 5, 3, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 6, 2, 2, 2, 4, 4, 2, 3, 2, 2, 5, 2, 2, 3, 4, 2, 2, 3, 2, 2, 3, 2, 7, 2, 3, 3, 2, 2, 2, 2, 3, 2, 2, 5, 4, 2, 3, 2, 2, 2, 2, 4, 3, 2, 3, 6, 2, 2, 2, 4, 2, 2, 5, 2, 3, 2, 2, 4, 2, 5, 2, 2, 3, 3, 8, 2, 2, 3, 2, 3, 4, 2, 2, 2, 3, 2, 2, 2, 2, 3, 3

Offset

1,1

Comments

The positive terms in A375339.

Links

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

Formula

a(n) = A051903(A190641(n)).

a(n) = A005361(A190641(n)).

a(n) = A375339(A190641(n)).

a(n) = A132349(A057521(A190641(n))).

a(n) = 2 if and only if A190641(n) is in A060687.

a(n) = 3 if and only if A190641(n) is in A048109.

a(n) <= 3 if and only if A190641(n) is in A082293.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (2*p-1)/((p-1)*(p^2-1)) / Sum_{p prime} 1/(p^2-1) = A375340 / A154945 = 2.74622231282166656595... .

Asymptotic second raw moment: <a^2> = Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)^2 = Sum_{p prime} (4*p^2-3*p+1)/((p-1)^3*(p+1)) / Sum_{p prime} 1/(p^2-1) = 9.064902009520365378603... .

Asymptotic second central moment, or variance, is <a^2> - <a>^2 = 1.52316501808078192104... and the asymptotic standard deviation is sqrt(<a^2> - <a>^2) = 1.23416571743051667098... .

Mathematica

s[n_] := Module[{e = Select[FactorInteger[n][[;; , 2]], # > 1 &]}, If[Length[e] == 1, e[[1]], Nothing]]; Array[s, 300]

Prog

(PARI) lista(kmax) = {my(e); for(k = 1, kmax, e = select(x -> x > 1, factor(k)[, 2]); if(#e == 1, print1(e[1], ", "))); }

Crossrefs

Cf. A005361, A048109, A051903, A060687, A082293, A154945, A190641, A375339, A375340.

Sequence in context: A212174 A375342 A368713 * A368039 A160558 A241019

Adjacent sequences: A375338 A375339 A375340 * A375342 A375343 A375344

Keyword

nonn,easy

Author

Amiram Eldar, Aug 12 2024

Status

approved