A120349 Refactorable numbers k such that the number of odd divisors r is odd, the number of even divisors s is even and both r and s are divisors of k.

36, 3600, 8100, 10000, 22500, 26244, 32400, 90000, 142884, 202500, 396900, 518400, 656100, 810000, 980100, 1285956, 1368900, 1587600, 1679616, 2286144, 2340900, 2624400, 2924100

Offset: 1

Author

Walter Kehowski, Jun 24 2006

Keywords

• nonn

Comments

In general, since n is even, r is always a multiple of s and even if both r and s are divisors of n, the sum t=r+s may not be. For example, if n=144, then r=3, s=12 and t=r+s=15.

Examples

			a(1)=36 since r=3(odd), s=6(even) and t=r+s=9 are all divisors.
		

Links

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

Crossrefs

Cf. A033950, A049439, A057265.

Programs

Maple

with(numtheory); T := proc(n::posint) local x, y, S; S:=divisors(n); x:=nops( select(z->type(z,odd),S) ); y:=nops( select(z->type(z,even),S) ); return [x,y] end; RF:=[]: N:=12^6/2: CNT:=12^4: for w to 1 do for k from 1 to N do n:=2*k; if k mod CNT = 0 then print((N-k)/CNT) fi; r:=T(n)[1]; s:=T(n)[2]; t:=r+s; if type(s,even) and type(r,odd) and andmap(z -> n mod z = 0, [r,s,t]) then RF:=[op(RF),n]; print(n,r,s,t); fi; od od; RF;

Formula

a(n) = n-th number such that n is even, r = number of odd divisors of n, s = number of even divisors of n, t = r+s = number of divisors of n, are all divisors of n and r is odd, s is even.