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A375341
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The maximum exponent in the prime factorization of the numbers that have exactly one non-unitary prime factor.
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4
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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
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OFFSET
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1,1
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COMMENTS
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LINKS
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FORMULA
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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... .
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MATHEMATICA
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s[n_] := Module[{e = Select[FactorInteger[n][[;; , 2]], # > 1 &]}, If[Length[e] == 1, e[[1]], Nothing]]; Array[s, 300]
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PROG
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(PARI) lista(kmax) = {my(e); for(k = 1, kmax, e = select(x -> x > 1, factor(k)[, 2]); if(#e == 1, print1(e[1], ", "))); }
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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