A368473 Product of exponents of prime factorization of the exponentially 2^n-numbers (A138302).

1, 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, 4, 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

The terms of A005361 that are powers of 2 (A000079).

The first position of 2^k, for k = 0, 1, ..., is 1, 4, 15, 126, 1134, ..., which is the position of A085629(2^k) in A138302.

Links

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

Formula

a(n) = A005361(A138302(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/d) * Product_{p prime} (1 + Sum_{k>=0} 2^k/p^(2^k)) = 1.504710204899266020302..., where d = A271727 is the asymptotic density of the exponentially 2^n-numbers.

Mathematica

f[n_] := Module[{p = Times @@ FactorInteger[n][[;; , 2]]}, If[p == 2^IntegerExponent[p, 2], p, Nothing]]; Array[f, 150]

Prog

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

Crossrefs

Cf. A000079, A005361, A085629, A138302, A271727.

Cf. A366538, A367169.

Similar sequences: A322327, A368472, A368474.

Sequence in context: A259154 A369933 A374327 * A106035 A293811 A362228

Adjacent sequences: A368470 A368471 A368472 * A368474 A368475 A368476

Keyword

nonn,easy

Author

Amiram Eldar, Dec 26 2023

Status

approved