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%I #13 Jul 17 2015 12:30:28
%S 50,77,179,346,347,550,1758,1909,9205,27884,30660,37354,52019,88052,
%T 107974,131590,164413,232447,295682,326133,328491,1494561,1541005,
%U 1541851
%N Numbers n with the property that it is possible to write the base 2 expansion of n as concat(a_2,b_2), with a_2>0 and b_2>0 such that, converting a_2 and b_2 to base 10 as a and b, we have antisigma(a) + antisigma(b) = n.
%e 50 in base 2 is 110010. If we take 110010 = concat(1100,10) then 1100 and 10 converted to base 10 are 12 and 2. Finally 12*13/2 - sigma(12) + 2*3/2 - sigma(2) = 78 - 28 + 3 - 3 = 50.
%e 179 in base 2 is 1001101. If we take 1001101 = concat(11100,1) then 11100 and 1 converted to base 10 are 5 and 19. Finally 5*6/2 - sigma(5) + 19*20/2 - sigma(19) = 15 - 6 + 190 - 20 = 179.
%p with(numtheory): P:=proc(q) local a,b,c,j,k,n;
%p for n from 1 to q do c:=convert(n,binary,decimal);
%p j:=0; for k from 1 to ilog10(c) do
%p a:=convert(trunc(c/10^k),decimal,binary);
%p b:=convert((c mod 10^k),decimal,binary);
%p if a*b>0 then if a*(a+1)/2-sigma(a)+b*(b+1)/2-sigma(b)=n then print(n);
%p break; fi; fi; od; od; end: P(10^9);
%t f[n_] := Block[{d = IntegerDigits[n, 2], len = IntegerLength[n, 2], k}, ReplaceAll[Reap[Do[k = {FromDigits[Take[d, i], 2], FromDigits[Take[d, -(len - i)], 2]}; If[! MemberQ[k, 0], Sow@ k], {i, 1, len - 1}]], {} -> {1}][[-1, 1]]]; Select[Range@ 100000, MemberQ[Total /@ (# (# + 1)/2 - DivisorSigma[1, #] &) /@ f@ #, #] &] (* _Michael De Vlieger_, Jul 03 2015 *)
%Y Cf. A024816, A253824, A253825, A258813, A258843, A258844.
%K nonn,base,more
%O 1,1
%A _Paolo P. Lava_, Jul 03 2015
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