A253825 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.

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

1,1

Comments

a(28) > 10^11. - Hiroaki Yamanouchi, Sep 26 2015

Links

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..27

Example

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.

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);

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 *)

Prog

(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

Crossrefs

Cf. A001065, A000203, A253824.

Sequence in context: A278187 A206125 A322935 * A186068 A186064 A251454

Adjacent sequences: A253822 A253823 A253824 * A253826 A253827 A253828

Keyword

nonn,base

Author

Paolo P. Lava, 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

Status

approved