A372466 The maximal exponent in the prime factorization of the numbers whose number of divisors is a power of 2 (A036537).

0, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3

Offset: 1

Author

Amiram Eldar, May 01 2024

Keywords

• nonn
• easy

Comments

All the terms are of the form 2^k-1 (A000225).

Links

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

Crossrefs

Cf. A000225, A036537, A051903, A327839, A372467.

Similar sequences: A368710, A368711, A368712, A368713, A369933, A369935.

Programs

Mathematica

pow2Q[n_] := n == 2^IntegerExponent[n, 2]; f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, pow2Q[# + 1] &], Max @@ e, Nothing]]; f[1] = 0; Array[f, 150]

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(apply(x -> x + 1, e))), print1(vecmax(e), ", "))); }

Formula

a(n) = A051903(A036537(n)).

a(n) = 2^A372467(n) - 1.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (d(1) + Sum_{k>=2} (2^k-1) * (d(k) - d(k-1))) / A327839 = 1.25306367526166810834..., where d(k) = Product_{p prime} (1 - 1/p + Sum_{i=1..k} (1/p^(2^i-1)-1/p^(2^i))).