A269135 Numbers n which are neither a prime nor a square of a prime such that there is no d, 2<=d<=n/2, which divides binomial(n-d-1,d-1) and is not coprime to n.

1, 6, 8, 10, 12, 15, 20, 21, 24, 33, 35, 143, 323, 899, 1763, 3599, 5183, 10403, 11663, 19043, 22499, 32399, 36863, 39203, 51983, 57599, 72899, 79523, 97343, 121103, 176399, 186623, 213443, 272483, 324899, 359999

Offset: 1

Author

Vladimir Shevelev, Feb 20 2016

Keywords

• nonn

Comments

Note that every d>1 divides binomial(n-d-1,d-1), if gcd(n,d)=1.

Theorem: A number m > 33 is a member if and only if it is a product p*(p+2), where p is lesser of twin primes (A001359).

This follows from Theorem 1 of the Shevelev (2007) link.

Links

• R. J. Mathar, Corrigendum to "On the divisibility of ...", arXiv:1109.0922 [math.NT], 2011.
• V. Shevelev, On divisibility of binomial(n-i-1,i-1) by i, Intl. J. of Number Theory 3, no.1 (2007), 119-139.

Crossrefs

Cf. A178071, A178098, A178099.

Programs

Mathematica

selQ[n_] := !PrimeQ[n] && !PrimeQ[Sqrt[n]] && NoneTrue[Range[2, n/2], Divisible[Binomial[n - # - 1, # - 1], #] && !CoprimeQ[n, #]&];
pp = Select[Prime[Range[200]], PrimeQ[# + 2] &];
Join[Select[Range[33], selQ], pp (pp + 2) // Rest] (* Jean-François Alcover, Sep 28 2018, after Shevelev's theorem *)

PARI

isok(n) = { if (!isprime(n) && !(issquare(n, &p) && isprime(p)), for (d=2, n\2, if ((gcd(n,d)!=1) && !(binomial(n-d-1,d-1) % d), return (0))); return (1););} \\ Michel Marcus, Feb 20 2016

Extensions

Typos in data corrected by Jean-François Alcover, Sep 28 2018