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%I #20 Sep 24 2015 11:26:22
%S 216,13296,13464,14416,51480,235200,575484,578592,585000,1032656,
%T 1121400,1599552,4190364,4786110,8365968,11268688,13010634,13253436,
%U 21835624,22108784,23896320,136311840,152820243,160380496,170073324,295999900,421686580,445421664
%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) = n.
%C a(31) > 5*10^8. - _Hiroaki Yamanouchi_, Sep 24 2015
%H Hiroaki Yamanouchi, <a href="/A259831/b259831.txt">Table of n, a(n) for n = 1..30</a>
%e 216 in base 2 is 11011000. If we take 11011000 = concat(110,11000) then 110 and 11000 converted to base 10 are 6 and 24. Finally (sigma(6) - 6)*(sigma(24) - 24) = (12 - 6)*(60 - 24) = 6 * 36 = 216;
%e 13296 in base 2 is 11001111110000. If we take 11001111110000 = concat(110,01111110000) then 110 and 01111110000 converted to base 10 are 6 and 1008. Finally (sigma(6) - 6)*(sigma(1008) - 1008) = (12 - 6)*(3224 - 1008)= 6 * 2216 = 13296.
%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 (sigma(a)-a)*(sigma(b)-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] - #1) (DivisorSigma[1, #2] - #2) & @@@ f@ #, #] &] (* _Michael De Vlieger_, Jul 07 2015 *)
%o (Python)
%o from sympy import divisor_sigma
%o A259831_list= []
%o for n in range(2,10**6):
%o ....s = format(n,'0b')
%o ....for l in range(1,len(s)):
%o ........n1, n2 = int(s[:l],2), int(s[l:],2)
%o ........if n2 > 0 and n == (divisor_sigma(n1)-n1)*(divisor_sigma(n2)-n2):
%o ............A259831_list.append(n)
%o ............break # _Chai Wah Wu_, Jul 17 2015
%Y Cf. A001065, A244079, A258813, A258843, A259832.
%K nonn,base
%O 1,1
%A _Paolo P. Lava_, Jul 06 2015
%E a(13)-a(14) from _Chai Wah Wu_, Jul 17 2015
%E a(15)-a(28) from _Hiroaki Yamanouchi_, Sep 24 2015
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