A396549 The sum of the coreful divisors of n that are not exponential unitary divisors of n.

0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 9, 0, 0, 0, 0, 28, 0, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 0, 18, 0, 28, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 0, 0, 60, 36, 0, 0, 0, 0, 0, 0, 44, 0, 0, 0, 0, 0, 0, 0, 84, 0, 0, 0, 0

Offset

1,8

Comments

All the exponential unitary divisors (A361255) of n are by definition coreful divisors (A284318) of n (see A396548).

The number of these divisors is A396548(n).

Links

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

Formula

a(n) = A057723(n) - A322857(n).

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

a(n) = A396544(n) if and only if n is an exponentially squarefree number (A209061).

Mathematica

f1[p_, e_] = (p^(e+1)-1)/(p-1) - 1;
f2[p_, e_] = DivisorSum[e, p^# &, CoprimeQ[#, e/#] &];
a[n_] := Module[{fct = FactorInteger[n]}, Times @@ f1 @@@ fct - Times @@ f2 @@@ fct]; a[1] = 0; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2]+1) - 1)/(f[i, 1]-1) - 1) - prod(i = 1, #f~, sumdiv(f[i, 2], d, if(gcd(d, f[i, 2]/d) == 1, f[i, 1]^d))); }

Crossrefs

Cf. A004709, A057723, A209061, A284318, A322857, A361255, A396544, A396548.

Sequence in context: A396544 A308429 A308428 * A308276 A389885 A340977

Adjacent sequences: A396546 A396547 A396548 * A396550 A396551 A396552

Keyword

nonn,easy

Author

Amiram Eldar, May 29 2026

Status

approved