A390868 The number of integers k from 1 to n such that gcd(n, k) is a non-bi-unitary divisor of n.

0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 3, 0, 0, 0, 2, 0, 4, 0, 5, 0, 0, 0, 0, 4, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 11, 10, 0, 0, 6, 6, 8, 0, 13, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 14, 4, 0, 0, 0, 17, 0, 0, 0, 16, 0, 0, 12, 19, 0, 0, 0, 10, 6, 0, 0, 21

Offset

1,9

Comments

A non-bi-unitary divisor d of n is a divisor d such that the greatest common unitary divisor of d and n/d is larger than 1.

Links

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

Formula

a(n) = n - A390867(n).

a(n) > 0 if and only if n is in A072587, i.e., n is not an exponentially odd number (A268335).

Dirichlet g.f.: zeta(s-1) * (1 - zeta(2*s-1) * Product_{p prime} (1 - 2/p^(2*s-1) + 1/p^(2*s) + 1/p^(3*s-2) - 1/p^(3*s-1))).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 - zeta(3) * Product_{p prime} (1 - 2/p^3 + 2/p^4 - 1/p^5) = 0.060581449926056231448... .

Mathematica

f[p_, e_] := p^e - If[EvenQ[e], (p - 1)*p^(e/2 - 1), 0]; a[1] = 0; a[n_] := n - Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); n - prod(i = 1, #f~, f[i, 1]^f[i, 2] - if(!(f[i, 2] % 2), (f[i, 1]-1)*f[i, 1]^(f[i, 2]/2 - 1))); }

Crossrefs

Cf. A002117, A072587, A165430, A222266, A268335.

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), A390865 (exponential), A390866 (nonexponential), A390867 (bi-unitary), this sequence (non-bi-unitary).

Sequence in context: A091731 A284269 A140579 * A132681 A127648 A212209

Adjacent sequences: A390865 A390866 A390867 * A390869 A390870 A390871

Keyword

nonn,easy

Author

Amiram Eldar, Nov 22 2025

Status

approved