A392629 The number of coreful infinitary divisors of n that are numbers whose number of divisors is a power of 2.

1, 1, 1, 0, 1, 1, 1, 2, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 2, 0, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 2, 1, 2, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 2, 1

Offset

1,8

Comments

For the definition of a coreful divisor see A284318 and A307958, and for the definition of an infinitary divisor see A037445.

Links

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

Formula

Multiplicative with a(p^e) = A007814(e+1).

a(n) = 0 if and only if n is not an exponentially odd number (A268335, i.e., n is in A072587).

a(n) <= A363329(n), with equality if and only if n is squarefree (A005117).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 0.79378624236775721631..., where f(x) = (1-x) * (1 + Sum_{k>=1} x^(2^k-1)/(1-x^(2^k))).

Mathematica

f[p_, e_] := IntegerExponent[e + 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = vecprod(apply(x -> valuation(x+1, 2), factor(n)[, 2]));

Crossrefs

Cf. A005117, A007814, A036537, A037445, A072587, A077609, A268335, A284318, A307958, A363329, A392628, A392630.

Sequence in context: A305490 A374060 A375149 * A113706 A279952 A054845

Adjacent sequences: A392626 A392627 A392628 * A392630 A392631 A392632

Keyword

nonn,mult,easy

Author

Amiram Eldar, Jan 18 2026

Status

approved