A324555 a(n) = the smallest number m such that gcd(tau(m), pod(m)) = n where tau(k) = the number of the divisors of k (A000005) and pod(k) = the product of the divisors of k (A007955).

1, 2, 9, 6, 400, 12, 3136, 24, 36, 80, 123904, 60, 692224, 448, 2025, 120, 18939904, 180, 94633984, 240, 35721, 11264, 2218786816, 360, 10000, 53248, 900, 1344, 225754218496, 720, 1031865892864, 840, 7144929, 1114112, 1960000, 1260, 94076963651584, 4980736

Offset: 1

Author

Jaroslav Krizek, Mar 05 2019

Keywords

• nonn

Comments

a(n) = the smallest number m such that A306671(m) = n.

If a(17) exists, it must be bigger than 10^7.

Examples

			For n=3; a(3) = 9 because gcd(tau(9), pod(9)) = gcd (3, 27) = 3 and 9 is the smallest.
		

Crossrefs

Cf. A000005, A007955, A306671.

Programs

Magma

[Min([n: n in[1..10^6] | GCD(NumberOfDivisors(n), &*[d: d in Divisors(n)]) eq k]): k in [1..16]]

Mathematica

Array[Block[{m = 1}, While[GCD[DivisorSigma[0, m], Times @@ Divisors@ m] != #, m++]; m] &, 16] (* Michael De Vlieger, Mar 24 2019 *)

PARI

a(n) = {my(k=1, vk = divisors(k)); while(gcd(#vk, vecprod(vk)) != n, k++; vk = divisors(k)); k;} \\ Michel Marcus, Mar 06 2019

Extensions

a(17)-a(38) from Jon E. Schoenfield, Mar 07 2019