A396548 - OEIS

%I #10 May 31 2026 09:44:57

%S 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,

%T 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,

%U 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

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

%C First differs from A380399 at n = 64.

%C 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.

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

%C The sum of these divisors is A396549(n).

%H Amiram Eldar, <a href="/A396548/b396548.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

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

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

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

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

%K nonn,easy

%O 1,16

%A _Amiram Eldar_, May 29 2026