A324322 - OEIS

%I #16 Jan 15 2023 15:09:10

%S 12,13,14,15,16,17,18,19,32,43,52,54,65,72,73,76,87,92,94,98,103,352,

%T 461,571,792,803,1003

%N Numbers k such that Ld(k) == k (mod Rd(k)), where Ld(k) = A067080 and Rd(k) = A067079.

%C No other term up to 2*10^10. - _Giovanni Resta_, Feb 22 2019

%e Ld(792) = 792*79*7 = 437976, Rd(792) = 792*92*2 = 145728 and 437976 == 792 (mod 145728).

%p with(numtheory): P:=proc(n) local a,k;

%p a:=mul(n mod 10^k, k=1..ilog10(n)+1): if a>0 then

%p if n=mul(trunc(n/10^k), k=0..ilog10(n)) mod a then n;

%p fi; fi; end: seq(P(i),i=1..1100);

%t Select[Range[10^6], If[#2 != 0, Mod[Times @@ Map[FromDigits, NestWhileList[Most@ # &, IntegerDigits@ #1, Length@ # > 1 &]], #2] == #1] & @@ {#, Times @@ Map[FromDigits, NestWhileList[Rest@ # &, IntegerDigits@ #, Length@ # > 1 &]]} &] (* _Michael De Vlieger_, Feb 25 2019 *)

%o (PARI) Ld(n) = my(t=n); while(n\=10, t*=n); t; \\ A067080

%o Rd(n) = prod(k=1, logint(n+!n, 10)+1, n-n\10^k*10^k); \\ A067079

%o isok(k) = if (k % 10, (Ld(k) % Rd(k)) == k); \\ _Michel Marcus_, Jan 15 2023

%Y Cf. A067079, A067080, A324322.

%K nonn,more

%O 1,1

%A _Paolo P. Lava_, Feb 22 2019