A110419 Least number k such that (prime(n)-1)! concatenated with k == 0 (mod prime(n)).

0, 1, 0, 3, 12, 22, 15, 24, 31, 13, 38, 26, 18, 14, 53, 47, 41, 39, 33, 29, 27, 21, 17, 11, 127, 192, 176, 144, 128, 209, 111, 214, 178, 166, 106, 245, 215, 185, 165, 135, 105, 276, 236, 228, 212, 204, 156, 108, 319, 313, 301, 283, 277, 247, 229, 211, 193, 187

Offset

1,4

Comments

From Robert Israel, Jun 16 2025: (Start)

a(n) is the least number k such that (prime(n)-1) concatenated with k == 0 (mod prime(n)).

a(n) is the least k >= 0 such that k == 10^d (mod prime(n)) where k has d decimal digits. (End)

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

a(5) = 12, 10! concatenated with 12 = 362880012 == 0 (mod prime(5)).

Maple

c0:=proc(x, y) local s: s:=proc(m) nops(convert(m, base, 10)) end: if y=0 then 10*x else x*10^s(y)+y: fi end: a:=proc(n) local p: p:=proc(k) if c0((ithprime(n)-1)!, k) mod ithprime(n) = 0 then k else fi end: [seq(p(k), k=0..400)][1] end: seq(a(n), n=1..75); # c0 yields the concatenation of two numbers # Emeric Deutsch, Aug 05 2005
# Alternative:
f:= proc(p) local d, k, j;
 for d from 1 + ilog10(p) do;
   k:= 10^d mod p;
   if k < 10^(d-1) then j:= ceil((10^(d-1)-k)/p); k:= k + j*p fi;
   if k < 10^d then return k fi;
 od;
end proc;
f(2):= 0: f(5):= 0:
map(f @ ithprime, [$1..100]); # Robert Israel, Jun 16 2025

Mathematica

Do[p = Prime[n]; k = 0; s = ToString[(p-1)! ]; While[Mod[ToExpression[s <> ToString[k]], p] > 0, k++ ]; Print[k], {n, 1, 50}] (* Ryan Propper, Aug 05 2005 *)
lnk[n_]:=Module[{p=Prime[n], c, k=0}, c=(Prime[n]-1)!; While[Mod[ c*10^ IntegerLength[ k]+k, p]!=0, k++]; k]; Join[{0, 1, 0}, Array[lnk, 60, 4]] (* Harvey P. Dale, Dec 27 2019 *)

Crossrefs

Cf. A110418, A110420.

Sequence in context: A051656 A074004 A088099 * A031223 A063598 A063107

Adjacent sequences: A110416 A110417 A110418 * A110420 A110421 A110422

Keyword

base,easy,nonn,look

Author

Amarnath Murthy, Aug 01 2005

Extensions

More terms from Ryan Propper and Emeric Deutsch, Aug 05 2005

Status

approved