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%I #22 Sep 24 2015 11:26:31
%S 7708,9020,86934,92128,120228,325180,372000,491630,565724,739032,
%T 862780,1120024,1344090,1419304,1440858,1678232,2752626,2980515,
%U 3684344,4154418,4860476,7539610,7565257,9527064,11025372,12277728,17002336,20256672,22528536,24597984
%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)-a)*(sigma(b)-b) = sigma(n).
%C a(62) > 10^8. - _Hiroaki Yamanouchi_, Sep 24 2015
%H Hiroaki Yamanouchi, <a href="/A259832/b259832.txt">Table of n, a(n) for n = 1..61</a>
%e 7708 in base 2 is 1111000011100. If we take 1111000011100 = concat(11110000, 11100) then 11110000 and 11100 converted to base 10 are 240 and 28. Finally (sigma(240) - 240)*(sigma(28) - 28) = (744 - 240)*(56 - 28) = 504 * 28 = 14112 = sigma(7708); 9020 in base 2 is 10001100111100. If we take 10001100111100= concat(10001100, 111100) then 110 and 01111110000 converted to base 10 are 140 and 60. Finally (sigma(140) - 140)*(sigma(60) - 60) = (336 - 140)*(168 - 60)= 196 * 108 = 21160 = sigma(9020).
%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)-a)*(sigma(b)-b)=sigma(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@ 125000, MemberQ[(DivisorSigma[1, #1] - #1) (DivisorSigma[1, #2] - #2) & @@@ f@ #, DivisorSigma[1, #]] &] (* _Michael De Vlieger_, Jul 07 2015 *)
%o (Python)
%o from sympy import divisor_sigma
%o A259832_list= []
%o for n in range(2,10**6):
%o ....s, k = format(n,'0b'), divisor_sigma(n)
%o ....for l in range(1,len(s)):
%o ........n1, n2 = int(s[:l],2), int(s[l:],2)
%o ........if n2 > 0 and k == (divisor_sigma(n1)-n1)*(divisor_sigma(n2)-n2):
%o ............A259832_list.append(n)
%o ............break # _Chai Wah Wu_, Jul 17 2015
%Y Cf. A000203, A244079, A258813, A258843, A259831.
%K nonn,base
%O 1,1
%A _Paolo P. Lava_, Jul 06 2015
%E a(16)-a(21) from _Chai Wah Wu_, Jul 17 2015
%E a(22)-a(30) from _Hiroaki Yamanouchi_, Sep 24 2015
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