A334729 a(n) = Product_{d|n} gcd(tau(d), sigma(d)).

1, 1, 2, 1, 2, 8, 2, 1, 2, 4, 2, 16, 2, 8, 16, 1, 2, 24, 2, 24, 16, 8, 2, 64, 2, 4, 8, 16, 2, 1024, 2, 3, 16, 4, 16, 48, 2, 8, 16, 48, 2, 2048, 2, 48, 96, 8, 2, 128, 6, 12, 16, 8, 2, 768, 16, 128, 16, 4, 2, 147456, 2, 8, 32, 3, 16, 2048, 2, 24, 16, 1024, 2

Offset: 1

Author

Jaroslav Krizek, May 09 2020

Keywords

• nonn

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

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A334491 (Product_{d|n} gcd(d, sigma(d))), A334579 (Sum_{d|n} gcd(tau(d), sigma(d))).

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

Programs

Magma

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

Maple

g:= proc(d) option remember; igcd(numtheory:-tau(d), numtheory:-sigma(d)) end proc:
f:= n -> mul(g(d), d = numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, May 11 2020

Mathematica

a[n_] := Product[GCD[DivisorSigma[0, d], DivisorSigma[1, d]], {d, Divisors[n]}]; Array[a, 100] (* Amiram Eldar, May 09 2020 *)

PARI

a(n) = my(d=divisors(n)); prod(k=1, #d, gcd(numdiv(d[k]), sigma(d[k]))); \\ Michel Marcus, May 09-11 2020

Formula

a(p) = 2 for p = odd primes (A065091).