A334579 a(n) = Sum_{d|n} gcd(tau(d), sigma(d)).

1, 2, 3, 3, 3, 8, 3, 4, 4, 6, 3, 11, 3, 8, 9, 5, 3, 12, 3, 13, 9, 8, 3, 16, 4, 6, 8, 11, 3, 24, 3, 8, 9, 6, 9, 16, 3, 8, 9, 16, 3, 26, 3, 15, 16, 8, 3, 19, 6, 10, 9, 9, 3, 24, 9, 20, 9, 6, 3, 45, 3, 8, 12, 9, 9, 26, 3, 13, 9, 24, 3, 24, 3, 6, 12, 11, 9, 24, 3

Offset: 1

Author

Jaroslav Krizek, May 06 2020

Keywords

• nonn

Comments

Inverse Möbius transform of A009205. - Antti Karttunen, May 19 2020

Examples

			a(6) = gcd(tau(1), sigma(1)) + gcd(tau(2), sigma(2)) + gcd(tau(3), sigma(3)) + gcd(tau(6), sigma(6)) = gcd(1, 1) + gcd(2, 3) + gcd(2, 4) + gcd(4, 12) = 1 + 1 + 2 + 4 = 8.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A322979 (Sum_{d|n} gcd(d, tau(d))), A334490 (Sum_{d|n} gcd(d, sigma(d))).

Cf. A000005 (tau(n)), A000203 (sigma(n)), A009205 (gcd(tau(n), sigma(n))).

Cf. A334729 (with product, instead of sum).

Programs

Magma

[&+[GCD(#Divisors(d), &+Divisors(d)): d in Divisors(n)]: n in [1..100]]

Mathematica

a[n_] := DivisorSum[n, GCD[DivisorSigma[0, #], DivisorSigma[1, #]] &]; Array[a, 100] (* Amiram Eldar, May 07 2020 *)

PARI

a(n) = sumdiv(n, d, gcd(numdiv(d), sigma(d))); \\ Michel Marcus, May 07 2020

Formula

a(p) = 3 for p = odd primes (A065091).