A337325 a(n) is the smallest number m such that gcd(tau(m), sigma(m), pod(m)) = 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, 10, 18, 6, 5000, 90, 66339, 30, 288, 3240, 10036224, 60, 582160384, 20412, 16200, 168, 49030215219, 612, 4637065216, 1520, 142884, 912384, 98881718827959, 420, 7543125, 479232, 14112, 5824, 26559758051835904, 104400, 25796647321600, 840, 491774976, 1268973568

Offset: 1

Author

Jaroslav Krizek, Aug 23 2020

Keywords

• nonn

Comments

p^(q-1) | a(q). If p != q then (p^(q-1) * q) | a(q) for some primes p and q. A similar idea can be used for nonprime q. - David A. Corneth, Aug 25 2020

Examples

			For n = 6; a(6) = 90 because 90 is the smallest number with gcd(tau(90), sigma(90), pod(90)) = gcd(12, 234, 531441000000) = 6.
		

Crossrefs

Cf. A336722 (gcd(tau(n), sigma(n), pod(n))).

Cf. A337324 (least m such that gcd(m,tau(m),sigma(m),pod(m)) = n).

Cf. A000005, A000203, A007955.

Programs

Magma

[Min([m: m in[1..10^5] | GCD([#Divisors(m), &+Divisors(m), &*Divisors(m)]) eq k]): k in [1..10]]

PARI

f(n) = my(d=divisors(factor(n))); gcd([#d, vecsum(d), vecprod(d)]);
a(n) = my(m=1); while (f(m) != n, m++); m; \\ Michel Marcus, Sep 21 2020

Extensions

a(11) and a(13) from Amiram Eldar, Aug 25 2020

More terms from Jinyuan Wang, Oct 03 2020