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A369933
The maximal exponent in the prime factorization of the exponentially 2^n numbers (A138302).
4
0, 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
OFFSET
1,4
COMMENTS
Differs from A368473 at n = 1, 32, 89, 126, 159, ... .
LINKS
FORMULA
a(n) = A051903(A138302(n)).
a(n) = 2^A369934(n), for n >= 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/zeta(2) + Sum_{k>=1} (2^k * (d(k) - d(k-1))) / A271727 = 1.40540547368932408503..., where d(k) = Product_{p prime} (1 - 1/p^3 + Sum_{i=2..k} (1/p^(2^i)-1/p^(2^i+1))) for k >= 1, and d(0) = 1/zeta(2).
MATHEMATICA
pow2Q[n_] := n == 2^IntegerExponent[n, 2]; f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, pow2Q], Max @@ e, Nothing]]; f[1] = 0; Array[f, 150]
PROG
(PARI) ispow2(n) = n >> valuation(n, 2) == 1;
lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = factor(k)[, 2]; if(ispow2(vecprod(e)), print1(vecmax(e), ", "))); }
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Feb 06 2024
STATUS
approved