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A375342
The maximum exponent in the prime factorization of the numbers whose powerful part is a power of a squarefree number that is larger than 1.
1
2, 3, 2, 2, 4, 2, 2, 3, 2, 3, 2, 5, 2, 3, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 6, 2, 2, 2, 4, 4, 2, 3, 2, 2, 5, 2, 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, 2, 3, 2, 3, 6, 2, 2, 2, 2, 4, 2, 3, 2, 5, 2, 2, 3, 2, 2, 4, 2, 5, 2, 2, 3, 3, 2, 8, 2, 2, 3, 2, 3, 4, 2, 2, 2, 3
OFFSET
1,1
LINKS
FORMULA
a(n) = A051903(A375142(n)).
a(n) = 2 if and only if A375142(n) is in A067259.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=2} k*d(k) / Sum_{k>=2} d(k) = 2.70113273169250927084..., where d(k) = (f(k)-1)/zeta(2) is the asymptotic density of terms m of A375142 with A051903(m) = k, f(k) = zeta(k) * Product_{p prime} (1 + Sum_{i=k+1..2*k-1} (-1)^i/p^i), if k is even, and f(k) = (zeta(2*k)/zeta(k)) * Product_{p prime} (1 + 2/p^k + Sum_{i=k+1..2*k-1} (-1)^(i+1)/p^i) if k is odd > 1.
MATHEMATICA
s[n_] := Module[{e = Select[FactorInteger[n][[;; , 2]], # > 1 &]}, If[Length[e] > 0 && SameQ @@ e, e[[1]], Nothing]]; Array[s, 300]
PROG
(PARI) lista(kmax) = {my(e); for(k = 1, kmax, e = select(x -> x > 1, factor(k)[, 2]); if(#e > 0 && vecmin(e) == vecmax(e), print1(e[1], ", "))); }
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Aug 12 2024
STATUS
approved