A260684 - OEIS

A260684

Irregular triangular array read by rows. Row n gives the primes in the prime factorization of n! that have exponent of 1.

%I #10 Nov 18 2015 15:36:16

%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 p-Sylow subgroups in S_n is binomial(n,p)*(p-2)!. By Wilson's Theorem (p-2)!==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 7;

%e 7, 11;

%e 7, 11, 13;

%e 11, 13;

%e 11, 13, 17;

%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