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

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, 11, 1, 22

Offset

2,1

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

Links

Alois P. Heinz, Rows n = 2..157, flattened

Vyacheslav M. Abramov, A solution to a Paul Erdos problem, arXiv:2504.19392 [math.NT], 2025. [withdrawn by author]

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.

Example

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

Maple

T:= proc(n) option remember; local f, k, p, l; f, p:= 1, ithprime(n);
      l:=[][]; for k from 2 to p-1 do f:= f*k mod p; l:= l, f od: l
    end:
seq(T(n), n=2..10);  # Alois P. Heinz, May 31 2026

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 *)

Prog

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

Crossrefs

Cf. A062169.

T(n,n) gives A091858 (for n>=2).

Row sums give A371035.

Sequence in context: A098804 A191320 A180228 * A197376 A113072 A328025

Adjacent sequences: A212317 A212318 A212319 * A212321 A212322 A212323

Keyword

nonn,tabf

Author

Michel Marcus, Oct 25 2013

Status

approved