A336722 a(n) = gcd(tau(n), sigma(n), pod(n)) where tau(k) is the number of divisors of k (A000005), sigma(k) is the sum of divisors of k (A000203) and pod(k) is the product of divisors of k (A007955).

1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 4, 1, 1, 1, 3, 1, 2, 1, 4, 1, 4, 1, 2, 1, 2, 1, 8, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 8, 1, 2, 3, 4, 1, 2, 1, 1, 1, 2, 1, 8, 1, 8, 1, 2, 1, 12, 1, 4, 1, 1, 1, 8, 1, 2, 1, 8, 1, 3, 1, 2, 1, 2, 1, 8, 1, 2, 1, 2, 1, 4, 1, 4, 1, 4, 1, 6, 1, 2, 1, 4, 1, 12, 1, 1, 3, 1, 1, 8, 1, 2, 1

Offset: 1

Author

Jaroslav Krizek, Aug 01 2020

Keywords

• nonn

Comments

a(n) = tau(n) for numbers n: 1, 6, 14, 22, 30, 38, 42, 46, 54, 56, 60, 62, 66, 70, 78, 86, 94, 96, 102, ...

Examples

			a(6) = gcd(tau(6), sigma(6), pod(6)) = gcd(4, 12, 36) = 4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A009205 (gcd(tau(n), sigma(n))), A306671 (gcd(tau(n), pod(n))), A306682 (gcd(sigma(n), pod(n))).

Cf. A000005 (tau(n)), A000203 (sigma(n)), A007955 (pod(n)), A336723 (lcm(tau(n), sigma(n), pod(n))).

Cf. A277521 (numbers k such that a(k) = tau(k) and simultaneously A336723(k) = pod(k)).

Programs

Magma

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

Mathematica

a[n_] := GCD @@ {(d = DivisorSigma[0,n]), DivisorSigma[1, n], n^(d/2)}; Array[a, 100] (* Amiram Eldar, Aug 01 2020 *)

PARI

A007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2)); \\ From A007955
A336722(n) = gcd(A007955(n), gcd(numdiv(n), sigma(n))); \\ Antti Karttunen, Aug 10 2020

Formula

a(p) = 1 for p = primes (A000040).

a(n) = gcd(A007955(n), A009205(n)). - Antti Karttunen, Aug 10 2020

Extensions

Data section extended up to a(105) by Antti Karttunen, Aug 10 2020