A343717 a(n) is the smallest number that yields a prime when appended to n!.

1, 1, 3, 1, 1, 1, 7, 11, 29, 17, 43, 29, 13, 47, 19, 73, 37, 19, 41, 103, 41, 31, 43, 1, 113, 31, 37, 59, 41, 53, 41, 47, 1, 41, 149, 37, 53, 73, 337, 1, 103, 151, 293, 47, 107, 509, 127, 71, 167, 197, 167, 149, 67, 163, 139, 251, 59, 107, 241, 331, 269, 1, 149

Offset

0,3

Comments

Appending to n! any number k <= n yields a multiple of k; that multiple cannot be prime except at k=1, so, for every n, a(n)=1 or a(n) > n.

a(n) = 1 iff n = 0 or n is in A024912.

See A068695 for a proof that a(n) always exists. - Felix Fröhlich, May 18 2021

If a(n) is composite, then a(n) > 2n. - Michael S. Branicky, May 18 2021

Links

Lucas A. Brown, Table of n, a(n) for n = 0..2630 (terms 0..1000 from Michael S. Branicky).

Michael S. Branicky, Python program for b-file and A343718, A343719

Formula

a(n) = A068695(n!) = A068695(A000142(n)).

Example

n=1: 1! = 1; appending a 1 yields 11, a prime, so a(1)=1.
n=2: 2! = 2; appending a 1 yields 21 = 3*7, and appending a 2 yields 22 = 2*11, but appending a 3 yields 23 (a prime), so a(2)=3.
n=19: 19! = 121645100408832000; appending any number < 103 yields a composite, but 121645100408832000103 is a prime, so a(19)=103.

Maple

a:= proc(n) option remember; local k, t; t:= n!;
      for k while not isprime(parse(cat(t, k))) do od; k
    end:
seq(a(n), n=0..62);  # Alois P. Heinz, May 17 2021

Mathematica

Array[Block[{m = #!, k = 0}, While[! PrimeQ[10^If[k == 0, 1, IntegerLength[k]]*m + k], k++]; k] &, 62] (* Michael De Vlieger, May 17 2021 *)
snp[n_]:=Module[{nf=n!, c=1}, While[!PrimeQ[nf*10^IntegerLength[c]+c], c++]; c]; Array[snp, 70, 0] (* Harvey P. Dale, Oct 17 2024 *)

Prog

(Python) # see link for faster program producing b-file
from sympy import factorial, isprime
def a(n):
  start = str(factorial(n))
  end = 1
  while not isprime(int(start + str(end))): end += 2
  return end
print([a(n) for n in range(63)]) # Michael S. Branicky, May 17 2021
(PARI) for(n=0, 62, my(f=digits(n!)); forstep(k=1, oo, 2, my(p=fromdigits(concat(f, digits(k)))); if(ispseudoprime(p), print1(k, ", "); break))) \\ Hugo Pfoertner, May 18 2021

Crossrefs

Cf. A000040, A000142, A024912, A033932, A068695, A095194, A343718, A343719.

Sequence in context: A362897 A005765 A360289 * A263159 A229142 A156535

Adjacent sequences: A343714 A343715 A343716 * A343718 A343719 A343720

Keyword

nonn,base

Author

Jon E. Schoenfield, May 17 2021

Status

approved