%I
%S 2,2,3,3,3,5,5,5,7,5,7,5,7,7,7,11,7,11,7,11,13,11,13,11,13,11,13,11,
%T 13,17,11,13,17,11,13,17,19,11,13,17,19,11,13,17,19,13,17,19,13,17,19,
%U 23,13,17,19,23,13,17,19,23
%N Irregular triangular array read by rows. Row n gives the primes in the prime factorization of n! that have exponent of 1.
%C For any prime p in row n, binomial(n,p)==1 (mod p). This is a consequence of Sylow's (3rd) Theorem. For these primes the number of pSylow subgroups in S_n is binomial(n,p)*(p2)!. By Wilson's Theorem (p2)!==1 (mod p) so that binomial(n,p)==1 (mod p).
%H Alois P. Heinz, <a href="/A260684/b260684.txt">Rows n = 2..500, flattened</a>
%e 2;
%e 2, 3;
%e 3;
%e 3, 5;
%e 5;
%e 5, 7;
%e 5, 7;
%e 5, 7;
%e 7;
%e 7, 11;
%e 7, 11;
%e 7, 11, 13;
%e 11, 13;
%e 11, 13;
%e 11, 13;
%e 11, 13, 17;
%e 11, 13, 17;
%e 11, 13, 17, 19;
%e 11, 13, 17, 19;
%t Table[Select[FactorInteger[n!], #[[2]] == 1 &][[All, 1]], {n, 2,20}] // Grid
%Y Cf. A000142.
%Y The last entry in each row gives A007917.
%K nonn,tabf
%O 2,1
%A _Geoffrey Critzer_, Nov 15 2015
