

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



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
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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



