A120359 - OEIS

A120359

Even refactorable numbers k such that the number r of odd divisors is odd, the number s of even divisors is even, both r and s are divisors of k and k is the first number for which the triple (r,s,t) occurs, where t is the number of divisors of k.

%I #11 Aug 02 2024 01:55:28

%S 36,3600,8100,10000,26244,32400,142884,202500,396900,518400,656100,

%T 810000,1587600,1679616,2286144,2624400,3572100,6350400,9144576,

%U 9922500,12960000,14288400,20575296,25401600,28579716,32148900,39690000,41990400,48024900,57153600,89302500

%N Even refactorable numbers k such that the number r of odd divisors is odd, the number s of even divisors is even, both r and s are divisors of k and k is the first number for which the triple (r,s,t) occurs, where t is the number of divisors of k.

%C Note that s is necessarily a multiple of r.

%e a(1) = 36 since r = 3, s = 6 and t = r+s = 9.

%t seq[kmax_] := Module[{triples = {}, v = {}, r, s, t}, Do[t = DivisorSigma[0, k]; r = t - DivisorSigma[0, k/2]; s = t - r; If[OddQ[r] && EvenQ[s] && FreeQ[triples, {r, s, t}] && Divisible[k, t] && Divisible[k, r] && Divisible[k, s], AppendTo[v, k]; AppendTo[triples, {r, s, t}]], {k, 2, kmax, 2}]; v]; seq[10^6] (* _Amiram Eldar_, Aug 01 2024 *)

%Y Cf. A000005, A001227, A033950, A049439, A057265, A120358, A120361, A183063.

%K nonn

%O 1,1

%A _Walter Kehowski_, Jun 25 2006

%E a(21)-a(22) inserted and a(24)-a(31) added by _Amiram Eldar_, Aug 01 2024