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%I #14 Jan 15 2020 05:50:05
%S 2,12,18,24,36,72,80,180,240,252,360,396,450,468,480,504,560,612,684,
%T 720,792,828,880,882,896,936,972,1040,1044,1116,1200,1224,1250,1332,
%U 1344,1360,1368,1440,1476,1520,1548,1620,1656,1692,1840,1908,1944,2000
%N Refactorable numbers k such that the number of odd divisors and the number of even divisors of k are both divisors of k.
%C Since s = 0 if k is odd, the number k is necessarily even and then the number of even divisors s is always a multiple of the number of odd divisors r. Note that t = r + s may not be a divisor even if both r and s are divisors. For example, if k = 144, then r = 3, s = 12, but t = r + s = 15.
%H Amiram Eldar, <a href="/A120350/b120350.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = k is even, r = number of odd divisors of k, s = number of even divisors of k and t = r + s = number of divisors of k, are all divisors of k.
%e a(3) = 18 since r = 3, s = 3 and t = r + s = 6 are all divisors.
%p with(numtheory); A:=[]: for w to 1 do for k from 1 to 5000 do n:=2*k; S:=divisors(n); r:=nops( select(z->type(z,odd),S) ); s:=nops( select(z->type(z,even),S) ); t:=r+s; if andmap(z -> n mod z = 0, [r,s,t]) then A:=[op(A),n]; print(n,r,s,t); fi; od od; A;
%t oddtau[n_] := DivisorSigma[0, n/2^IntegerExponent[n, 2]]; seqQ[n_] := Module[{d = DivisorSigma[0, n], o = odd[n]}, Divisible[n, d] && Divisible[n, o] && Divisible[n, d - o]]; Select[Range[2, 2000, 2], seqQ] (* _Amiram Eldar_, Jan 15 2020 *)
%Y Cf. A033950, A049439, A057265.
%K nonn
%O 1,1
%A _Walter Kehowski_, Jun 24 2006
%E Offset corrected by _Amiram Eldar_, Jan 15 2020
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