A363329 a(n) is the number of divisors of n that are both coreful and infinitary.

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, May 28 2023

Keywords

• nonn
• mult

Comments

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

If e > 0 is the exponent of the highest power of p dividing n (where p is a prime), then for each divisor d of n that is both a coreful and an infinitary divisor, the exponent of the highest power of p dividing d is a number k >= 1 such that the bitwise AND of e and k is equal to k.

The least term that does not equal 1 or 3 is a(128) = 7.

The range of this sequence is A282572.

Examples

			a(8) = 3 since 8 has 4 divisors, 1, 2, 4 and 8, all are infinitary and 3 of them (2, 4 and 8) are also coreful.
		

Links

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

Crossrefs

Cf. A000120, A005361 (number of coreful divisors), A007947, A037445, A077609, A138302, A282572, A359411.

Programs

Mathematica

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

PARI

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

Python

from math import prod
from sympy import factorint
def A363329(n): return prod((1<Chai Wah Wu, Sep 01 2023

Formula

Multiplicative with a(p^e) = 2^A000120(e) - 1.

a(n) = 1 is and only if n is in A138302.

a(n) >= A359411(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (-1/p + (1-1/p)*Product_{k>=0} (1 + 2/p^(2^k))) = 1.29926464312956239535... .