A382392 a(n) is the least prime number whose factorial base expansion contains the digit n.

2, 2, 5, 19, 97, 601, 4327, 35281, 322571, 3265949, 36288017, 439084817, 5748019201, 80951270459, 1220496076831, 19615115520037, 334764638208037, 6046686277632071, 115242726703104073, 2311256907767808001, 48658040163532800037, 1072909785605898240031

Offset: 0

Author

Rémy Sigrist, Mar 23 2025

Keywords

• nonn
• base

Comments

This sequence is well defined: a(0) = a(1) = 2, and for n > 1, (n+1)! and n*n! + 1 are coprime, so by Dirichlet's theorem on arithmetic progressions, there exists a prime number p of the form k*(n+1)! + n*n! + 1 for some k >= 0, and the factorial base expansion of this prime number contains the digit n, hence a(n) <= p.

Examples

			The initial terms, in decimal and in factorial base, are:
  n  a(n)     fact(a(n))
  -  -------  -----------------
  0        2  1,0
  1        2  1,0
  2        5  2,1
  3       19  3,0,1
  4       97  4,0,0,1
  5      601  5,0,0,0,1
  6     4327  6,0,0,1,0,1
  7    35281  7,0,0,0,0,0,1
  8   322571  8,0,0,0,0,1,2,1
  9  3265949  9,0,0,0,0,1,0,2,1
		

Links

• Index entries for sequences related to factorial base representation

Crossrefs

Cf. A001563, A062584, A090703.

Programs

PARI

a(n) = { forprime (p = n*n!, oo, my (q = p); for (r = 2, oo, if (q==0, break, q % r==n, return (p), q \= r););); }

Formula

a(n) > A001563(n).