A390867 The number of integers k from 1 to n such that gcd(n, k) is a bi-unitary divisor of n.

1, 2, 3, 3, 5, 6, 7, 8, 7, 10, 11, 9, 13, 14, 15, 14, 17, 14, 19, 15, 21, 22, 23, 24, 21, 26, 27, 21, 29, 30, 31, 32, 33, 34, 35, 21, 37, 38, 39, 40, 41, 42, 43, 33, 35, 46, 47, 42, 43, 42, 51, 39, 53, 54, 55, 56, 57, 58, 59, 45, 61, 62, 49, 60, 65, 66, 67, 51, 69

Offset

1,2

Comments

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

Links

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

Formula

Multiplicative with a(p^e) = p^e if e is odd, and a(p^e) = p^e - (p-1)*p^(e/2 - 1) if e is even.

a(n) = n - A390868(n).

a(n) <= n, with equality if and only if n is an exponentially odd number (A268335).

Dirichlet g.f.: zeta(s-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 = zeta(3) * Product_{p prime} (1 - 2/p^3 + 2/p^4 - 1/p^5) = 0.93941855007394376855... .

Mathematica

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

Prog

(PARI) a(n) = {my(f = factor(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, 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), this sequence (bi-unitary), A390868 (non-bi-unitary).

Sequence in context: A384041 A384048 A390863 * A384054 A334819 A338375

Adjacent sequences: A390864 A390865 A390866 * A390868 A390869 A390870

Keyword

nonn,mult,easy

Author

Amiram Eldar, Nov 22 2025

Status

approved