A334621 Number of unitary prime divisors, p, of n such that n-p is squarefree.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 2, 1, 1, 0, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 2, 1, 0, 0, 0, 1, 0, 1, 2, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1

Offset

1,33

Links

Table of n, a(n) for n=1..96.

Formula

a(n) = Sum_{p|n, p prime, gcd(p,n/p) = 1} mu(n-p)^2, where mu is the Moebius function (A008683).

Example

a(10) = 1; 5 is a unitary prime divisor of 10 since gcd(5,2) = 1, and 10 - 5 = 5 (squarefree).
a(33) = 2; 3 is a unitary prime divisor of 33 since gcd(3,11) = 1, and 33 - 3 = 30 (squarefree).

Mathematica

Table[Sum[MoebiusMu[n - i]^2*KroneckerDelta[GCD[i, n/i], 1] (PrimePi[i] - PrimePi[i - 1]) (1 - Ceiling[n/i] + Floor[n/i]), {i, n}], {n, 100}]

Crossrefs

Cf. A005117 (squarefree numbers), A008683 (Moebius), A056169 (number of unitary prime divisors of n).

Sequence in context: A204179 A204244 A083093 * A293899 A015794 A011650

Adjacent sequences: A334618 A334619 A334620 * A334622 A334623 A334624

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Sep 09 2020

Status

approved