Amiram Eldar, <a href="/A374327/b374327_1.txt">Table of n, a(n) for n = 1..10000</a>

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Amiram Eldar, <a href="/A374327/b374327_1.txt">Table of n, a(n) for n = 1..10000</a>

Similar sequences: A374324, A374325, A374326, A374328.

allocatedThe maximal exponent in the prime factorization of the numbers whose maximal exponent in their prime factorization is a forpower Amiramof Eldar2.

1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2

1,3

First differs from {A369933(n+1), n>=1} at n = 378.

The first occurrence of 2^k, for k = 0, 1, ..., is at 1, 3, 14, 224, 57307, ..., which is the position of 2^(2^k) at A369938.

a(n) = 2^A374328(n).

a(n) = A051903(A369938(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} 2^k * d(k) / Sum_{k>=0} d(k) = 1.41151462942556759486..., where d(k) = 1/zeta(2^k+1) - 1/zeta(2^k) for k>=1, and d(0) = 1/zeta(2).

f[n_] := Module[{e = If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]}, If[e == 2^IntegerExponent[e, 2], e, Nothing]]; Array[f, 150]

(PARI) lista(kmax) = {my(e); for(k = 2, kmax, e = vecmax(factor(k)[, 2]); if(e >> valuation(e, 2) == 1, print1(e, ", "))); }

Cf. A051903, A369933, A369938.

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Amiram Eldar, Jul 04 2024

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