A120351 Even numbers k such that the number of odd divisors r and the number of even divisors s are both divisors of k.

2, 4, 6, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 34, 36, 38, 44, 46, 48, 52, 58, 62, 68, 72, 74, 76, 80, 82, 86, 90, 92, 94, 106, 112, 116, 118, 120, 122, 124, 126, 134, 142, 144, 146, 148, 150, 158, 160, 164, 166, 168, 172, 176, 178, 180, 188, 192, 194, 198, 202, 206

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 s is always a multiple of 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

			16 is a term since r=1 and s=4 are both divisors.
		

Links

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

Crossrefs

Cf. A033950, A049439, A057265.

Programs

Maple

with(numtheory); A:=[]: N:=10^4/2: for w to 1 do for k from 2 to N do n:=2*k; S:=divisors(n); r:=nops( select(z->type(z,odd),S) ); s:=nops( select(z->type(z,even),S) ); if andmap(z -> n mod z = 0,[r,s]) then A:=[op(A),n]; print(n,r,s); fi; od od; A;

Mathematica

aQ[n_] := Divisible[n, (ev = DivisorSigma[0, n/2])] && Divisible[n, DivisorSigma[0, n] - ev]; Select[Range[2, 206, 2], aQ] (* Amiram Eldar, Nov 02 2019 *)

Formula

a(n) = n is even, r = number of odd divisors of n, s = number of even divisors of n, are all divisors of n.

Extensions

Term 2 inserted by Amiram Eldar, Nov 02 2019