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%I #13 Sep 26 2015 11:50:06
%S 6396,20680,124416,567816,1719480,7593432,10538040,36382320,107277800,
%T 123251968,166601760,327844840,933363000,1286859804,2524125184,
%U 3398418000,4561432920,4566915540,4911440776,7097433536,16913792670,20565608940,21099997800,27639552000
%N Numbers n = concat(s,t) such that n = (sigma(s)-s) * (sigma(t)-t), where sigma(x)-x is the sum of the aliquot parts of x.
%C a(28) > 10^11. - _Hiroaki Yamanouchi_, Sep 26 2015
%H Hiroaki Yamanouchi, <a href="/A253825/b253825.txt">Table of n, a(n) for n = 1..27</a>
%e 6396 = concat(63,96) -> sigma(63)-63 = 41, sigma(96)-96 = 156 and 41*156 = 6396.
%e 20680 = concat(20,680) -> sigma(20)-20 = 22, sigma(680)-680 = 940 and 22*940 = 20680.
%e 124416 = concat(12,4416) -> sigma(12)-12 = 16, sigma(4416)-4416 = 7776 and 16*7776 = 124416.
%e 567816 = concat(567,816) -> sigma(567)-567 = 410, sigma(816)-816 = 1416 and 401*1416 = 567816.
%p with(numtheory): P:=proc(q) local s, t, k, n;
%p for n from 1 to q do for k from 1 to ilog10(n) do s:=n mod 10^k; t:=trunc(n/10^k); if s*t>0 then if (sigma(s)-s)*(sigma(t)-t)=n
%p then print(n); break; fi; fi; od; od; end: P(10^6);
%t fQ[n_] := Block[{idn = IntegerDigits@ n, lng = Floor@ Log10@ n}, MemberQ[ Table[s = FromDigits@ Take[idn, {1, i}]; t = FromDigits@ Take[idn, {i + 1, lng + 1}]; (DivisorSigma[1, s] - s) (DivisorSigma[1, t] - t), {i, lng}], n]]; k = 1; lst = {}; While[k < 100000001, If[fQ@ k, AppendTo[lst, k]; Print@ k]; k++] (* _Robert G. Wilson v_, Jan 26 2015 *)
%o (PARI) isok(n) = {len = #Str(n); for (k=1, len-1, na = n\10^k; nb = n % 10^k; if (nb && (n == (sigma(na)-na)*(sigma(nb)-nb)), return (1)););} \\ _Michel Marcus_, Jan 15 2015
%Y Cf. A001065, A000203, A253824.
%K nonn,base
%O 1,1
%A _Paolo P. Lava_, Jan 15 2015
%E a(8) & a(9) from _Robert G. Wilson v_, Jan 26 2015
%E a(10)-a(24) from _Hiroaki Yamanouchi_, Sep 26 2015
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