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.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

Cf. A051903, A057521, A067259, A375142, A375341.

Sequence in context: A174329 A295312 A212174 * A368713 A375341 A368039

Adjacent sequences: A375339 A375340 A375341 * A375343 A375344 A375345

Keyword

nonn,easy

Author

Amiram Eldar, Aug 12 2024

Status

approved