A392013 The number of exponential divisors of the exponentially-2^n numbers.

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 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, 3, 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, 3, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 4

Offset

1,4

Comments

The sum of these divisors is A392015(n).

Links

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

Formula

a(n) = A049419(A138302(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/d) * Product_{p prime} f(1/p) = 1.43886949217580662745..., where d = A271727, and f(x) = (1-x) * (1 + Sum_{k>=0} (k+1)*x^(2^k)).

Mathematica

f[p_, e_] := Module[{v = IntegerExponent[e, 2]}, If[e == 2^v, v + 1, 0]]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 100] , # > 0 &]

Prog

(PARI) s(n) = {my(f = factor(n)); prod(i = 1, #f~, my(v = valuation(f[i, 2], 2)); if(f[i, 2] == 1 << v, v+1, 0)); }
list(lim) = select(x -> x > 0, vector(lim, i, s(i)));

Crossrefs

Cf. A049419, A138302, A271727, A322791, A392014, A392015.

Sequence in context: A305393 A392014 A259154 * A369933 A374327 A368473

Adjacent sequences: A392010 A392011 A392012 * A392014 A392015 A392016

Keyword

nonn,easy

Author

Amiram Eldar, Dec 27 2025

Status

approved