A345266 a(n) = Sum_{p|n, p prime} gcd(p,n/p).

0, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 2, 2, 1, 4, 1, 3, 2, 2, 1, 3, 5, 2, 3, 3, 1, 3, 1, 2, 2, 2, 2, 5, 1, 2, 2, 3, 1, 3, 1, 3, 4, 2, 1, 3, 7, 6, 2, 3, 1, 4, 2, 3, 2, 2, 1, 4, 1, 2, 4, 2, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 6, 3, 2, 3, 1, 3, 3, 2, 1, 4, 2, 2, 2, 3, 1, 5, 2, 3, 2, 2, 2, 3, 1, 8, 4, 7, 1, 3, 1, 3, 3

Offset: 1

Author

Wesley Ivan Hurt, Jun 13 2021

Keywords

• nonn
• look

Examples

			a(18) = Sum_{p|18} gcd(p,18/p) = gcd(2,9) + gcd(3,6) = 1 + 3 = 4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to gcd's.

Crossrefs

Cf. A001221 (omega), A007947 (rad), A008472 (sopf), A345302.

Cf. A055155, A056169, A063958.

Programs

Mathematica

Table[Sum[GCD[k, n/k] (PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]

PARI

a(n) = my(f=factor(n), p); sum(k=1, #f~, p=f[k, 1]; gcd(p,n/p)); \\ Michel Marcus, Jun 16 2021

PARI

A345266(n) = vecsum(apply(p->gcd(p,n/p), factor(n)[,1])); \\ Antti Karttunen, Nov 13 2021

Formula

a(p) = 1 for p prime.

From Wesley Ivan Hurt, Nov 21 2021: (Start)

a(n) = A056169(n) + A063958(n).

If n is squarefree, then a(n) = omega(n).

a(p^k) = p for primes p and k >= 2. (End)

Extensions

Data section extended up to 105 terms by Antti Karttunen, Nov 13 2021