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Even refactorable numbers k such that the number r of odd divisors and the number s of even divisors are both odd 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.
2

%I #12 Aug 02 2024 01:55:23

%S 2,18,72,450,1250,4050,16200,52488,56448,64800,71442,101250,198450,

%T 235298,285768,328050,405000,793800,1036800,1312200,1620000,1786050,

%U 3175200,4572288,4961250,5248800,7144200,12700800,14289858,15059072,16074450,19845000,24012450,25920000,28576800

%N Even refactorable numbers k such that the number r of odd divisors and the number s of even divisors are both odd 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(2) = 18 since r = 3, s = 3 and t = r+s = 6.

%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] && OddQ[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, A120359, A120361, A183063.

%K nonn

%O 1,1

%A _Walter Kehowski_, Jun 25 2006

%E a(27)-a(35) from _Amiram Eldar_, Aug 01 2024