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

1,2

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

Table of n, a(n) for n=1..36.

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.

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 *)

Prog

(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

Crossrefs

Cf. A178071, A178098, A178099.

Sequence in context: A303580 A003663 A075396 * A369666 A092121 A005525

Adjacent sequences: A269132 A269133 A269134 * A269136 A269137 A269138

Keyword

nonn

Author

Vladimir Shevelev, Feb 20 2016

Extensions

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

Status

approved