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%I #10 Oct 14 2014 11:17:33
%S 54,436,2014,2466,3365,4143,4965,7922,9332,15426,17554,24006,32874,
%T 33574,39476,44296,49976,54118,83726,116174,137635,163964,164824,
%U 177546,203514,220789,235434,379096,420716,476475,597741,600354,604986,680266,736306,748966
%N Numbers n such that if n = a U b (where U denotes concatenation) then abs(sigma*(a) - sigma*(b)) = abs(sigma*(n) - n), where sigma*(n) is the sum of the anti-divisors of n.
%C Neither a or b minor than 2 are considered because numbers 1 and 2 have no anti-divisors.
%C Similar to A239563 but using anti-divisors instead of divisors.
%e Anti-divisors of 4143 are 2, 5, 6, 1657, 2762 and their sum is 4432. Consider 4143 as 4 U 143. Anti-divisors of 4 is 3 and of 143 are 2, 3, 5, 7, 15, 19, 22, 26, 41, 57, 95 whose sum is 292. At the end we have that 4432 - 4143 = 289 = 292 - 3.
%p with(numtheory);
%p T:=proc(t) local w, x, y; x:=t; y:=0; while x>0 do x:=trunc(x/10); y:=y+1; od; end:
%p P:=proc(q) local a, b, c, d, f, g, i, j, k,n;
%p for n from 1 to q do b:=T(n); k:=0; j:=n; while j mod 2<>1 do k:=k+1; j:=j/2; od;
%p a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
%p for i from 1 to b-1 do c:=trunc(n/10^i); d:=n-c*10^i; if c>2 and d>2 then
%p k:=0; j:=c; while j mod 2<>1 do k:=k+1; j:=j/2; od;
%p f:=sigma(2*c+1)+sigma(2*c-1)+sigma(c/2^k)*2^(k+1)-6*c-2;
%p k:=0; j:=d; while j mod 2<>1 do k:=k+1; j:=j/2; od;
%p g:=sigma(2*d+1)+sigma(2*d-1)+sigma(d/2^k)*2^(k+1)-6*d-2;
%p if abs(f-g)=abs(a-n) then print(n); break; fi; fi; od; od; end: P(10^9);
%Y Cf. A066272, A066417, A239686.
%K nonn,base
%O 1,1
%A _Paolo P. Lava_, Mar 24 2014
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