A396543 The number of coreful divisors of n that are not exponential divisors of n.

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

Offset

1,32

Comments

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 divides the p-adic valuation of n, then d is an exponential divisor of n. Therefore, all the exponential 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 A396544(n).

Links

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

Formula

a(n) = A005361(n) - A049419(n).

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

a(p^e) = A049820(e) for a prime p and e >= 1.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A082695 - A327837 = 0.341279334515341152707... .

Mathematica

a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Times @@ e - Times @@ DivisorSigma[0, e]]; Array[a, 100]

Prog

(PARI) a(n) = my(e = factor(n)[, 2]); vecprod(e) - vecprod(apply(numdiv, e));

Crossrefs

Cf. A004709, A005361, A049419, A049820, A082695, A118914, A284318, A322791, A327837, A396544.

Sequence in context: A171063 A308229 A318673 * A151795 A375199 A008437

Adjacent sequences: A396539 A396540 A396541 * A396544 A396545 A396546

Keyword

nonn,easy

Author

Amiram Eldar, May 29 2026

Status

approved