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%I #17 Jul 17 2015 12:30:01
%S 28,120,162,196,868,1260,1302,1800,2016,2480,3780,4464,6804,7440,8370,
%T 9882,22200,32640,34290,35640,40640,73152,127008,187488,213776,489888,
%U 572880,602640,674082,1074528,1077120,1397088,1536192,1582560,1662120,1669164,1781136,1905120
%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 sigma(a)*sigma(b) = n.
%e 28 in base 2 is 11100. If we take 11100 = concat(11,100) then 11 and 100 converted to base 2 are 3 and 4. Finally sigma(3)*sigma(4) = 4 * 7 = 28;
%e 120 in base 2 is 1111000. If we take 1111000 = concat(111,1000) then 111 and 1000 converted to base 10 are 7 and 8. Finally sigma(7)*sigma(8) = 8 * 15 = 120.
%p with(numtheory): P:=proc(q) local a,b,c,k,n;
%p for n from 1 to q do c:=convert(n, binary, decimal);
%p 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 sigma(a)*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[DivisorSigma[1, #1] DivisorSigma[1, #2] & @@@ f@ #, #] &] (* _Michael De Vlieger_, Jul 07 2015 *)
%Y Cf. A000203, A258813, A258843, A259831, A259832.
%K nonn,base
%O 1,1
%A _Paolo P. Lava_, Jul 06 2015
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