A324554 a(n) = the smallest number m such that gcd(tau(m), sigma(m)) = n where tau(k) = the number of the divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).

1, 3, 18, 6, 648, 20, 2916, 30, 288, 304, 82944, 60, 36864, 832, 16200, 168, 5509980288, 612, 31719424, 432, 23328, 44032, 247669456896, 420, 9487368, 258048, 14112, 2496, 31581162962944, 4176, 26843545600, 840, 4064256, 4390912, 42693156, 1980, 151801324109824

Offset: 1

Author

Jaroslav Krizek, Mar 05 2019

Keywords

• nonn

Comments

a(n) = the smallest number m such that A009205(m) = n.

a(p) = q^(c*p-1) * k for p prime, where q is some prime, c and k are positive integers. - David A. Corneth, Mar 07 2019

Examples

			For n=3; a(3) = 18 because gcd(tau(18), sigma(18)) = gcd (6, 39) = 3 and 18 is the smallest.
		

Crossrefs

Cf. A000005, A007955, A009205.

Cf. also A324553, A324555.

Programs

Magma

[Min([n: n in[1..10^5] | GCD(NumberOfDivisors(n), SumOfDivisors(n)) eq k]): k in [1..16]]

Mathematica

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

PARI

A324554search_and_print(searchlimit) = { my(m = Map(), k); for(n=1,searchlimit,k=gcd(sigma(n),numdiv(n)); if(!mapisdefined(m,k), mapput(m,k,n))); for(k=1, oo, if(!mapisdefined(m,k), break, print1(mapget(m,k), ", "))); }; \\ Antti Karttunen, Mar 06 2019

Extensions

a(17)-a(37) from Jon E. Schoenfield, Mar 06 2019