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

1,8

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.

Links

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

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... .

Example

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.

Mathematica

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

Prog

(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<<e.bit_count())-1 for e in factorint(n).values()) # Chai Wah Wu, Sep 01 2023

Crossrefs

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

Sequence in context: A326538 A333844 A317933 * A370079 A318497 A202150

Adjacent sequences: A363326 A363327 A363328 * A363330 A363331 A363332

Keyword

nonn,mult

Author

Amiram Eldar, May 28 2023

Status

approved