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A375339 If n has exactly one non-unitary prime factor then a(n) is the exponent of the highest power of this prime that divides n, otherwise a(n) = 0.

%I A375339 #14 Apr 26 2025 20:13:02
%S A375339 0,0,0,2,0,0,0,3,2,0,0,2,0,0,0,4,0,2,0,2,0,0,0,3,2,0,3,2,0,0,0,5,0,0,
%T A375339 0,0,0,0,0,3,0,0,0,2,2,0,0,4,2,2,0,2,0,3,0,3,0,0,0,2,0,0,2,6,0,0,0,2,
%U A375339 0,0,0,0,0,0,2,2,0,0,0,4,4,0,0,2,0,0,0,3,0,2,0,2,0,0,0,5,0,2,2,0,0,0,0,3,0
%N A375339 If n has exactly one non-unitary prime factor then a(n) is the exponent of the highest power of this prime that divides n, otherwise a(n) = 0.
%C A375339 First differs from A212172, A275812 and A372603 at n = 36.
%C A375339 If n = m * p^e, such that m is squarefree, p is a prime that does not divide m and e >= 2, then a(n) = e, otherwise a(n) = 0.
%C A375339 By definition all the positive terms are larger than 1.
%C A375339 The asymptotic density of 0's in this sequence is 1 - Sum_{p prime} (1/(p^2-1)) / zeta(2) = 1 - A059956 * A154945 = 0.66461069244308962639... .
%C A375339 The asymptotic density of the occurrences of k >= 2 in this sequence is Sum_{p prime} (1/(p^(k-1)*(p+1))) / zeta(2). E.g., 0.200755... (A271971) for k = 2, 0.0741777... for k = 3, and 0.0320652... for k = 4.
%H A375339 Amiram Eldar, <a href="/A375339/b375339.txt">Table of n, a(n) for n = 1..10000</a>
%F A375339 a(n) = A051903(n) * A359466(n).
%F A375339 a(n) = A005361(n) * A359466(n).
%F A375339 a(A190641(n)) >= 2.
%F A375339 a(n) = 2 if and only if n is in A060687.
%F A375339 a(n) = 3 if and only if n is in A048109.
%F A375339 a(n) <= 3 if and only if n is in A082293.
%F A375339 Asymptotic mean: <a> = Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (2*p-1)/((p-1)^2*(p+1)) / zeta(2) = A375340 / A013661 = 0.92105359989459565838... .
%F A375339 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)) / zeta(2) = 3.04027120804428071157... .
%F A375339 The asymptotic second central moment, or variance, is <a^2> - <a>^2 = 2.19193147416548680815... and the asymptotic standard deviation is sqrt(<a^2> - <a>^2) = 1.48051729951577627898... .
%t A375339 a[n_] := Module[{e = Select[FactorInteger[n][[;; , 2]], # > 1 &]}, If[Length[e] == 1, e[[1]], 0]]; Array[a, 100]
%o A375339 (PARI) a(n) = {my(e = select(x -> x > 1, factor(n)[,2])); if(#e == 1, e[1], 0);}
%Y A375339 Cf. A005361, A048109, A051903, A060687, A082293, A190641, A359466, A375341 (the positive terms).
%Y A375339 Cf. A013661, A059956, A154945, A271971, A375340.
%Y A375339 Cf. A212172, A275812, A372603.
%K A375339 nonn,easy
%O A375339 1,4
%A A375339 _Amiram Eldar_, Aug 12 2024