A120350 Refactorable numbers k such that the number of odd divisors and the number of even divisors of k are both divisors of k.

2, 12, 18, 24, 36, 72, 80, 180, 240, 252, 360, 396, 450, 468, 480, 504, 560, 612, 684, 720, 792, 828, 880, 882, 896, 936, 972, 1040, 1044, 1116, 1200, 1224, 1250, 1332, 1344, 1360, 1368, 1440, 1476, 1520, 1548, 1620, 1656, 1692, 1840, 1908, 1944, 2000

Offset: 1

Author

Walter Kehowski, Jun 24 2006

Keywords

• nonn

Comments

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.

Examples

			a(3) = 18 since r = 3, s = 3 and t = r + s = 6 are all divisors.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A033950, A049439, A057265.

Programs

Maple

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;

Mathematica

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 *)

Formula

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.

Extensions

Offset corrected by Amiram Eldar, Jan 15 2020