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A260684
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Irregular triangular array read by rows. Row n gives the primes in the prime factorization of n! that have exponent of 1.
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1
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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
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2,1
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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;
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MATHEMATICA
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Table[Select[FactorInteger[n!], #[[2]] == 1 &][[All, 1]], {n, 2, 20}] // Grid
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CROSSREFS
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The last entry in each row gives A007917.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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