A227977 Numbers n for which n = sigma*(x) = sigma*(y), where n = x + y and sigma*(n) is the sum of the anti-divisors of n.

154, 3136, 5536, 20066, 136036, 9550080, 78011830

Offset: 1

Author

Paolo P. Lava, Oct 07 2013

Keywords

• nonn

Comments

Up to a(7) the triples (n, x, y) are (154, 77, 77), (3136, 1568, 1568)(5536, 2768, 2768), (20066, 10368, 9698), (136036, 80753, 55283), (9550080, 4775040, 4775040), (78011830, 39348342, 38663488). - Giovanni Resta, Oct 08 2013

Examples

			n = 20066 = 9698 + 10368.
Anti-divisors of 9698 are 3, 4, 5, 7, 9, 15, 17, 45, 52, 119, 163, 431, 1141, 1293, 1492, 2155, 2771, 3879, 6465 and their sum is 20066 that is equal to n.
Anti-divisors of 10368 are 5, 11, 13, 29, 55, 65, 89, 143, 145, 233, 256, 319, 377, 715, 768, 1595, 1885, 2304, 4147, 6912 and their sum is 20066 that is equal to n.
		

Crossrefs

Cf. A066272, A066417, A210732.

Programs

Maple

with(numtheory); P:=proc(q) local a,b, i, j, k, n;
for n from 1 to q do for i from 1 to trunc(n/2) do
k:=0; j:=i; while j mod 2<>1 do k:=k+1; j:=j/2; od;
a:=sigma(2*i+1)+sigma(2*i-1)+sigma(i/2^k)*2^(k+1)-6*i-2;
k:=0; j:=n-i; while j mod 2<>1 do k:=k+1; j:=j/2; od;
b:=sigma(2*(n-i)+1)+sigma(2*(n-i)-1)+sigma((n-i)/2^k)*2^(k+1)-6*(n-i)-2;
if a=b and a=n then print(n); fi; od; od; end: P(10^6);

Extensions

a(5)-a(7) from Giovanni Resta, Oct 08 2013