A253825 Numbers m = concat(s,t) such that m = (sigma(s)-s) * (sigma(t)-t), where sigma(x)-x is the sum of the aliquot parts of x.
6396, 20680, 124416, 567816, 1719480, 7593432, 10538040, 36382320, 107277800, 123251968, 166601760, 327844840, 933363000, 1286859804, 2524125184, 3398418000, 4561432920, 4566915540, 4911440776, 7097433536, 16913792670, 20565608940, 21099997800, 27639552000
Offset: 1
Examples
6396 = concat(63,96) -> sigma(63)-63 = 41, sigma(96)-96 = 156 and 41*156 = 6396. 20680 = concat(20,680) -> sigma(20)-20 = 22, sigma(680)-680 = 940 and 22*940 = 20680. 124416 = concat(12,4416) -> sigma(12)-12 = 16, sigma(4416)-4416 = 7776 and 16*7776 = 124416. 567816 = concat(567,816) -> sigma(567)-567 = 410, sigma(816)-816 = 1416 and 401*1416 = 567816.
Links
- Max Alekseyev, Table of n, a(n) for n = 1..42 (first 27 terms from Hiroaki Yamanouchi)
Programs
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Maple
with(numtheory): P:=proc(q) local s, t, k, n; 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 then print(n); break; fi; fi; od; od; end: P(10^6);
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Mathematica
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 *) -
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
Extensions
a(8)-a(9) from Robert G. Wilson v, Jan 26 2015
a(10)-a(24) from Hiroaki Yamanouchi, Sep 26 2015