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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 (list; graph; refs; listen; history; text; internal format)
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 A301541 A240519

Adjacent sequences:  A260681 A260682 A260683 * A260685 A260686 A260687

KEYWORD

nonn,tabf

AUTHOR

Geoffrey Critzer, Nov 15 2015

STATUS

approved

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Last modified December 4 00:33 EST 2020. Contains 338920 sequences. (Running on oeis4.)