A390957 The number of non-bi-unitary divisors of n.

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 1, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 2, 2, 0, 0, 0, 2, 1, 0, 0, 4, 0, 0, 0, 0

Offset

1,12

Comments

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

The sum of these divisors is A319072(n).

Links

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

Formula

a(n) = A000005(n) - A286324(n).

a(n) = 0 if and only if n is an exponentially odd number (A268335).

Sum_{k=1..n} a(k) ~ (1-A)*n*(log(n) + 2*gamma - 1 - A*B/(1-A)), where gamma = A001620, A = A306071 and B = A306072.

Mathematica

a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Times @@ (e + 1) - Times @@ (If[OddQ[#], # + 1, #]& /@ e)]; a[1] = 0; Array[a, 100]

Prog

(PARI) a(n) = {my(e = factor(n)[, 2]); vecprod(apply(x -> x+1, e)) - vecprod(apply(x -> if(x % 2, x+1, x), e)); }

Crossrefs

Cf. A000005, A222266, A268335, A286324, A319072.

Cf. A001620, A306071, A306072.

Similar sequences: A048105, A160097, A348341.

Sequence in context: A354527 A341523 A079127 * A056674 A349798 A336107

Adjacent sequences: A390954 A390955 A390956 * A390958 A390959 A390960

Keyword

nonn,easy

Author

Amiram Eldar, Nov 25 2025

Status

approved