cp's OEIS Frontend

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.

%I A269135 #20 Sep 28 2018 03:06:55
%S A269135 1,6,8,10,12,15,20,21,24,33,35,143,323,899,1763,3599,5183,10403,11663,
%T A269135 19043,22499,32399,36863,39203,51983,57599,72899,79523,97343,121103,
%U A269135 176399,186623,213443,272483,324899,359999
%N 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.
%C A269135 Note that every d>1 divides binomial(n-d-1,d-1), if gcd(n,d)=1.
%C A269135 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).
%C A269135 This follows from Theorem 1 of the Shevelev (2007) link.
%H A269135 R. J. Mathar, <a href="http://arxiv.org/abs/1109.0922">Corrigendum to "On the divisibility of ..."</a>, arXiv:1109.0922 [math.NT], 2011.
%H A269135 V. Shevelev, <a href="http://dx.doi.org/10.1142/S179304210700078X">On divisibility of binomial(n-i-1,i-1) by i</a>, Intl. J. of Number Theory 3, no.1 (2007), 119-139.
%t A269135 selQ[n_] := !PrimeQ[n] && !PrimeQ[Sqrt[n]] && NoneTrue[Range[2, n/2], Divisible[Binomial[n - # - 1, # - 1], #] && !CoprimeQ[n, #]&];
%t A269135 pp = Select[Prime[Range[200]], PrimeQ[# + 2] &];
%t A269135 Join[Select[Range[33], selQ], pp (pp + 2) // Rest] (* _Jean-François Alcover_, Sep 28 2018, after Shevelev's theorem *)
%o A269135 (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
%Y A269135 Cf. A178071, A178098, A178099.
%K A269135 nonn
%O A269135 1,2
%A A269135 _Vladimir Shevelev_, Feb 20 2016
%E A269135 Typos in data corrected by _Jean-François Alcover_, Sep 28 2018