A260684 Irregular triangular array read by rows. Row n gives the primes in the prime factorization of n! that have exponent of 1.

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, 13, 17, 11, 13, 17, 11, 13, 17, 19, 11, 13, 17, 19, 11, 13, 17, 19, 13, 17, 19, 13, 17, 19, 23, 13, 17, 19, 23, 13, 17, 19, 23

Offset

2,1

Comments

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).

Links

Alois P. Heinz, Rows n = 2..500, flattened

Example

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, 13, 17;
11, 13, 17;
11, 13, 17, 19;
11, 13, 17, 19;

Mathematica

Table[Select[FactorInteger[n!], #[[2]] == 1 &][[All, 1]], {n, 2, 20}] // Grid

Crossrefs

Cf. A000142.

The last entry in each row gives A007917.

Sequence in context: A108035 A202503 A049747 * A029093 A369788 A301541

Adjacent sequences: A260681 A260682 A260683 * A260685 A260686 A260687

Keyword

nonn,tabf

Author

Geoffrey Critzer, Nov 15 2015

Status

approved