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

%I A375341 #7 Aug 12 2024 13:09:42
%S A375341 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,
%T A375341 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,
%U A375341 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
%N A375341 The maximum exponent in the prime factorization of the numbers that have exactly one non-unitary prime factor.
%C A375341 The positive terms in A375339.
%H A375341 Amiram Eldar, <a href="/A375341/b375341.txt">Table of n, a(n) for n = 1..10000</a>
%F A375341 a(n) = A051903(A190641(n)).
%F A375341 a(n) = A005361(A190641(n)).
%F A375341 a(n) = A375339(A190641(n)).
%F A375341 a(n) = A132349(A057521(A190641(n))).
%F A375341 a(n) = 2 if and only if A190641(n) is in A060687.
%F A375341 a(n) = 3 if and only if A190641(n) is in A048109.
%F A375341 a(n) <= 3 if and only if A190641(n) is in A082293.
%F A375341 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... .
%F A375341 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... .
%F A375341 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... .
%t A375341 s[n_] := Module[{e = Select[FactorInteger[n][[;; , 2]], # > 1 &]}, If[Length[e] == 1, e[[1]], Nothing]]; Array[s, 300]
%o A375341 (PARI) lista(kmax) = {my(e); for(k = 1, kmax, e = select(x -> x > 1, factor(k)[,2]); if(#e == 1, print1(e[1], ", ")));}
%Y A375341 Cf. A005361, A048109, A051903, A060687, A082293, A154945, A190641, A375339, A375340.
%K A375341 nonn,easy
%O A375341 1,1
%A A375341 _Amiram Eldar_, Aug 12 2024