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

1,1

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.

Links

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

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.

Example

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

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 *)

Crossrefs

Cf. A033950, A049439, A057265.

Sequence in context: A277961 A294998 A323762 * A293851 A032413 A262897

Adjacent sequences: A120347 A120348 A120349 * A120351 A120352 A120353

Keyword

nonn

Author

Walter Kehowski, Jun 24 2006

Extensions

Offset corrected by Amiram Eldar, Jan 15 2020

Status

approved