A212320 Irregular triangle: T(n, k) = k! modulo prime(n), 1

2, 2, 1, 4, 2, 6, 3, 1, 6, 2, 6, 2, 10, 5, 2, 5, 1, 10, 2, 6, 11, 3, 5, 9, 7, 11, 6, 1, 12, 2, 6, 7, 1, 6, 8, 13, 15, 14, 1, 12, 3, 8, 1, 16, 2, 6, 5, 6, 17, 5, 2, 18, 9, 4, 10, 16, 15, 16, 9, 1, 18, 2, 6, 1, 5, 7, 3, 1, 9, 21, 1, 12, 18, 22, 8, 13, 14, 22, 4

Offset: 2

Author

Michel Marcus, Oct 25 2013

Keywords

• nonn
• tabf

Comments

It is conjectured that only first and second row have all terms distinct.

This holds for n less than ten million. In Trudgian's terminology, there are no socialist primes less than 10^7. - Charles R Greathouse IV, Nov 05 2013

Examples

			Irregular triangle begins:
  2;
  2, 1, 4;
  2, 6, 3,  1, 6;
  2, 6, 2, 10, 5, 2, 5, 1, 10;
		

Links

• Vyacheslav M. Abramov, A solution to a Paul Erdos problem, arXiv:2504.19392 [math.NT], 2025.
• W. D. Banks, F. Luca, I. E. Shparlinski, and H. Stichtenoth, On the Value Set of n! Modulo a Prime, Turk. J. Math., 29, (2005), 169-174.
• B. Rokowska and A. Schinzel, Sur un problème de M. Erdős, Elem. Math., 15:84-85, 1960, MR117188 (22 #7970). [Broken link]
• Tim Trudgian, There are no socialist primes less than 10^6, arXiv:1310.6403 [math.NT], 2013.

Crossrefs

Cf. A062169.

Programs

Mathematica

row[n_] := With[{p = Prime[n]}, Mod[Range[2, p-1]!, p]]; Table[row[n], {n, 2, 9}] // Flatten (* Jean-François Alcover, Oct 25 2013 *)

PARI

row(n) = {p = prime(n); for (i = 2, p-1, print1(i! % p, ", ");); print();}