A244079 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.

28, 120, 162, 196, 868, 1260, 1302, 1800, 2016, 2480, 3780, 4464, 6804, 7440, 8370, 9882, 22200, 32640, 34290, 35640, 40640, 73152, 127008, 187488, 213776, 489888, 572880, 602640, 674082, 1074528, 1077120, 1397088, 1536192, 1582560, 1662120, 1669164, 1781136, 1905120

Offset: 1

Author

Paolo P. Lava, Jul 06 2015

Keywords

• nonn
• base

Examples

			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;
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.
		

Crossrefs

Cf. A000203, A258813, A258843, A259831, A259832.

Programs

Maple

with(numtheory): P:=proc(q) local a,b,c,k,n;
for n from 1 to q do c:=convert(n, binary, decimal);
for k from 1 to ilog10(c) do
a:=convert(trunc(c/10^k), decimal, binary);
b:=convert((c mod 10^k), decimal, binary);
if a*b>0 then if sigma(a)*sigma(b)=n then print(n);
break; fi; fi; od; od; end: P(10^9);

Mathematica

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 *)