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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.
0
154, 3136, 5536, 20066, 136036, 9550080, 78011830
OFFSET
1,1
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
EXAMPLE
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.
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);
CROSSREFS
KEYWORD
nonn
AUTHOR
Paolo P. Lava, Oct 07 2013
EXTENSIONS
a(5)-a(7) from Giovanni Resta, Oct 08 2013
STATUS
approved