

A260684


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


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
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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 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).


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 A301541 A240519
Adjacent sequences: A260681 A260682 A260683 * A260685 A260686 A260687


KEYWORD

nonn,tabf


AUTHOR

Geoffrey Critzer, Nov 15 2015


STATUS

approved



