A396548 The number of coreful divisors of n that are not exponential unitary divisors of n.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0

Offset

1,16

Comments

First differs from A380399 at n = 64.

The coreful divisors (A284318) of n are the divisors d of n such that every prime factor of n also divides d. If, additionally, for every prime p dividing n, the p-adic valuation (the exponent of p in the prime factorization) of d is a unitary divisor of the p-adic valuation of n, then d is an exponential unitary divisor of n. Therefore, all the exponential unitary divisors of n are by definition coreful divisors of n.

a(n) depends only on the prime signature of n (A118914).

The sum of these divisors is A396549(n).

Links

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

Formula

a(n) = A005361(n) - A278908(n).

a(n) = 0 if and only if n is cubefree (A004709).

a(n) = A396543(n) if and only if n is an exponentially squarefree number (A209061).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A082695 - A393489 = 0.400431117447579324113... .

Mathematica

a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Times @@ e - Times @@ (2^PrimeNu[e])]; Array[a, 100]

Prog

(PARI) a(n) = my(e = factor(n)[, 2]); vecprod(e) - vecprod(apply(e -> 1<<omega(e), e));

Crossrefs

Cf. A004709, A082695, A118914, A209061, A278908, A284318, A361255, A380399, A393489, A396543, A396549.

Sequence in context: A322583 A351439 A102354 * A380399 A370222 A372717

Adjacent sequences: A396545 A396546 A396547 * A396549 A396550 A396551

Keyword

nonn,easy

Author

Amiram Eldar, May 29 2026

Status

approved