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

Author

Amiram Eldar, Aug 12 2024

Keywords

• nonn
• easy

Comments

The positive terms in A375339.

Links

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

Crossrefs

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

Programs

Mathematica

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

PARI

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

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: = 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 - ^2 = 1.52316501808078192104... and the asymptotic standard deviation is sqrt( - ^2) = 1.23416571743051667098... .