OFFSET
1,4
COMMENTS
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.
By definition all the positive terms are larger than 1.
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... .
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 0.01474101... for k = 4.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
a(A190641(n)) >= 2.
a(n) = 2 if and only if n is in A060687.
a(n) = 3 if and only if n is in A048109.
a(n) <= 3 if and only if n is in A082293.
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... .
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... .
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... .
MATHEMATICA
a[n_] := Module[{e = Select[FactorInteger[n][[;; , 2]], # > 1 &]}, If[Length[e] == 1, e[[1]], 0]]; Array[a, 100]
PROG
(PARI) a(n) = {my(e = select(x -> x > 1, factor(n)[, 2])); if(#e == 1, e[1], 0); }
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Aug 12 2024
STATUS
approved