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%I #42 Jul 26 2017 17:21:37
%S 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,
%T 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,
%U 18,2,6,1,5,7,3,1,9,21,1,12,18,22,8,13,14,22,4
%N Irregular triangle: T(n, k) = k! modulo prime(n), 1<k<prime(n), 1<n.
%C It is conjectured that only first and second row have all terms distinct.
%C 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
%H W. D. Banks, F. Luca, I. E. Shparlinski, H. Stichtenoth, <a href="http://journals.tubitak.gov.tr/math/abstract.htm?id=7510">On the Value Set of n! Modulo a Prime</a>, Turk. J. Math., 29, (2005), 169-174.
%H B. Rokowska and A. Schinzel, <a href="http://retro.seals.ch/digbib/view?pid=elemat-001:1960:15::144">Sur un problème de M. Erdős</a>, Elem. Math., 15:84-85, 1960, MR117188 (22 #7970). [Broken link]
%H Tim Trudgian, <a href="http://arxiv.org/abs/1310.6403">There are no socialist primes less than 10^6</a>, arXiv:1310.6403 [math.NT], 2013.
%e Irregular triangle begins:
%e 2;
%e 2, 1, 4;
%e 2, 6, 3, 1, 6;
%e 2, 6, 2, 10, 5, 2, 5, 1, 10;
%t 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 *)
%o (PARI) row(n) = {p = prime(n); for (i = 2, p-1, print1(i! % p, ", ");); print();}
%Y Cf. A062169.
%K nonn,tabf
%O 2,1
%A _Michel Marcus_, Oct 25 2013
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