A383865 The number of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or an infinitary divisor of the p-adic valuation of n.

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 3, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 3, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 6, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 5, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 6, 3, 4, 2, 12, 4, 4, 4

Offset: 1

Author

Amiram Eldar, May 12 2025

Keywords

• nonn
• easy
• mult

Comments

First differs from A383863 at n = 256.

The number of divisors d of n such that each is a unitary divisor of an exponential infinitary divisor of n (see A383760).

Analogous to the number of (1+e)-divisors (A049599) as exponential infinitary divisors (A383760, A307848) are analogous to exponential divisors (A322791, A049419).

The sum of these divisors is A383866(n).

Links

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

Crossrefs

Cf. A036537, A037445, A049419, A049599, A307848, A322791, A383760, A383863, A383866.

Programs

Mathematica

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

PARI

d(n) = vecprod(apply(x -> 2^hammingweight(x), factor(n)[, 2]));
a(n) = vecprod(apply(x -> 1 + d(x), factor(n)[, 2]));

Formula

Multiplicative with a(p^e) = 1 + A037445(e).

a(n) <= A049599(n), with equality if and only if all the exponents in the prime factorization of n are in A036537.