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A276710 Composite numbers m such that Product_{k=0..m} binomial(m,k) is divisible by m^(m-1). 1
36, 40, 63, 80, 84, 90, 105, 108, 132, 144, 150, 154, 160, 165, 168, 175, 176, 180, 182, 195, 198, 200, 208, 210, 220, 260, 264, 270, 273, 275, 280, 286, 288, 297, 300, 306, 308, 312, 315, 320, 324, 330, 340, 351, 357, 360, 364, 374, 378, 380, 385, 390 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The numbers Product_{k=0..m}binomial(m,k) form the sequence A001142(m). When m is a prime, the m-1 factors for 0<k<m are all divisible by m and therefore A001142(m) is divisible by m^(m-1). When m is a composite, this is generally not so, except for the numbers listed here (a variety of pseudoprimes).

Conjecture, tested so far up to m = 3828: "When m belongs to this list, Product_{k=0..m} binomial(m,k) is divisible also by m^m". Since this is impossible for prime m (see, e.g., A109874), the conjecture is equivalent to the statement "m is prime if and only if Product_{k=0..m} binomial(m,k) is divisible by m^(m-1) but not by m^m".

LINKS

Stanislav Sykora and Chai Wah Wu, Table of n, a(n) for n = 1..10000, terms for n = 1..797 from Stanislav Sykora

EXAMPLE

Since 36 is composite and 36^35 divides Product_{k=1..36}binomial(36,k), 36 is a member. In addition, it turns out that 36^36 also divides the product.

MATHEMATICA

Select[DeleteCases[Range[2, 400], p_ /; PrimeQ@ p], Divisible[Product[Binomial[#, k], {k, 0, #}], #^(# - 1)] &] (* Michael De Vlieger, Sep 16 2016 *)

PROG

(PARI) generator() = { /* Operates on two pre-defined integer vectors a, b of the same size. a(n) receives the terms of this sequence, while b(n) receives 0 if n^n|Product(binomial(n, k)), or 1 otherwise, and serves exclusively to test the conjecture. */

  my (m=1, n=0, p); for(k=1, #a, a[k]=0; b[k]=0);

  while(1, m++; p=prod(k=1, m-1, binomial(m, k));

    if((p%m^(m-1)==0)&&(!isprime(m)), n++; a[n]=m;

    if(p%m^m==0, b[n]=0, b[n]=1); if(n==#a, break)));

}

  a=vector(1000); b=a; generator();

  a /* Displays the result.

  Note: execution was interrupted due to excessive execution time */

CROSSREFS

Cf. A000169 (n^(n-1)), A001142, A109873, A109874.

Sequence in context: A077090 A254836 A067672 * A181484 A060292 A261265

Adjacent sequences:  A276707 A276708 A276709 * A276711 A276712 A276713

KEYWORD

nonn

AUTHOR

Stanislav Sykora, Sep 15 2016

STATUS

approved

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Last modified July 25 20:28 EDT 2017. Contains 289797 sequences.