A388973 The number of exponential divisors of the numbers whose number of exponential divisors is a power of 2.

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

Offset

1,4

Links

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

Formula

a(n) = A049419(A388972(n)).

a(n) = 2^A388974(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/d) * Product_{p prime} ((1 - 1/p) * (1 + Sum_{k>=1} A000005(A036537(k))/p^k)) = 1.50562990102073270647..., where d = 0.957956... is the asymptotic density of A388972 (see the Formula section of A388972).

Mathematica

f[p_, e_] := DivisorSigma[0, e]; d[1] = 1;
d[n_] := Times @@ f @@@ FactorInteger[n];
pow2Q[n_] := n == 2^IntegerExponent[n, 2];
q[n_] := pow2Q[d[n]]; d /@ Select[Range[100], q]

Prog

(PARI) d(n) = vecprod(apply(numdiv, factor(n)[, 2])); \\ A049419
ispow2(n) = n >> valuation(n, 2) == 1;
list(kmax) = {my(d1); for(k = 1, kmax, d1 = d(k); if(ispow2(d1), print1(d1, ", "))); }

Crossrefs

Cf. A000005, A036537, A049419, A372468, A388972, A388974.

Sequence in context: A366309 A392450 A394425 * A391988 A007424 A369163

Adjacent sequences: A388970 A388971 A388972 * A388974 A388975 A388976

Keyword

nonn,easy

Author

Amiram Eldar, Sep 22 2025

Status

approved