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URL: https://oeis.org/A390865

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A390865
The number of integers k from 1 to n such that gcd(n, k) is an exponential divisor of n.
6
1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1, 7, 1, 3, 1, 2, 1, 1, 1, 3, 5, 1, 7, 2, 1, 1, 1, 9, 1, 1, 1, 6, 1, 1, 1, 3, 1, 1, 1, 2, 3, 1, 1, 7, 7, 5, 1, 2, 1, 7, 1, 3, 1, 1, 1, 2, 1, 1, 3, 29, 1, 1, 1, 2, 1, 1, 1, 9, 1, 1, 5, 2, 1, 1, 1, 7, 25, 1, 1, 2, 1, 1, 1
OFFSET
1,4
LINKS
FORMULA
Multiplicative with a(p^e) = 1 + (p-1) * Sum_{d|e, d<e} p^(e-d-1).
a(n) = n - A390866(n).
a(n) <= A000010(n), with equality if and only if n = 1, 2, or 4.
a(n) = 1 if and only if n is squarefree (A005117).
MATHEMATICA
f[p_, e_] := 1 + (p-1) * DivisorSum[e, p^(e-#-1) &, # < e &]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + (f[i, 1]-1) * sumdiv(f[i, 2], d, (d < f[i, 2]) * f[i, 1]^(f[i, 2] - d -1))); }
CROSSREFS
The number of integers k from 1 to n such that gcd(n, k) is a divisor of n of type: A003557 (coreful), A055653 (unitary), A055654 (nonunitary), A010848 (non-coreful), A390863 (infinitary), A390864 (noninfinitary), this sequence (exponential), A390866 (nonexponential), A390867 (bi-unitary), A390868 (non-bi-unitary).
Sequence in context: A152798 A079115 A072906 * A239062 A341052 A201160
KEYWORD
nonn,mult,easy
AUTHOR
Amiram Eldar, Nov 22 2025
STATUS
approved