A210732 Numbers n for which sigma*(n)=sigma*(x)+sigma*(y), where n=x+y and sigma*(n) is the sum of the anti-divisors of n.

6, 9, 15, 18, 21, 24, 27, 30, 31, 33, 37, 39, 43, 44, 46, 47, 53, 56, 57, 62, 65, 66, 70, 73, 74, 75, 76, 78, 81, 83, 86, 88, 90, 91, 92, 93, 97, 99, 102, 103, 106, 107, 109, 110, 114, 116, 117, 118, 119, 121, 122, 123, 125, 126, 127, 129, 131, 133, 135, 136

Offset: 3

Author

Paolo P. Lava, May 10 2012

Keywords

• nonn

Comments

Similar to A211223 but using anti-divisors.

Examples

			sigma*(127)=sigma*(45)+sigma*(82) that is 212=86+126.
In more than one way:
sigma*(133)=sigma*(50)+sigma*(83)=sigma*(52)+sigma*(81) that is
204=80+124=94+110.
		

Crossrefs

Cf. A066272, A066417, A083207, A204830, A204831, A211223-A211225.

Programs

Maple

with(numtheory);
A210732:=proc(q)
local a,b,c,i,j,k,n;
for n from 3 to q do
  a:=0;
  for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then a:=a+k; fi; od;
  for i from 1 to trunc(n/2) do
   b:=0; c:=0;
   for k from 2 to i-1 do if abs((i mod k)-k/2)<1 then b:=b+k; fi; od;
   for k from 2 to n-i-1 do if abs(((n-i) mod k)-k/2)<1 then c:=c+k; fi; od;
   if a=b+c then print(n); break; fi;
  od;
od; end:
A210732(10000);