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

0,1

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.

Links

Table of n, a(n) for n=0..21.

Index entries for sequences related to factorial base representation

Formula

a(n) > A001563(n).

Example

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

Prog

(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); ); ); }

Crossrefs

Cf. A001563, A062584, A090703.

Sequence in context: A158059 A173323 A019099 * A384274 A154647 A103890

Adjacent sequences: A382389 A382390 A382391 * A382393 A382394 A382395

Keyword

nonn,base

Author

Rémy Sigrist, Mar 23 2025

Status

approved