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A106639 Distinguished primes.

%I A106639 #23 Mar 05 2015 14:38:57
%S A106639 2,3,5,7,11,13,19,23,29,37,43,59,61,67,83,157,173,227,277,283,317,347,
%T A106639 563,653,733,787,877,907,997,1213,1237,1283,1307,1523,1867,2083,2693,
%U A106639 2797,2803,3253,3413,3517,3643,3677,3733,3803,4253,4363,4547,4723,5387
%N A106639 Distinguished primes.
%C A106639 Primes are distinguished among the integers by having the fewest possible divisors. Among the primes, which primes are similarly distinguished? The distinguished primes have the fewest possible divisors in the neighborhood. Specifically, p is a distinguished prime iff together p-1, p and p+1, have 7 or fewer prime factors, counting multiple factors. Of course, the definition could be adjusted to make 3, or even 2, the unique distinguished prime, but then the sequence of distinguished primes would be severely truncated.
%C A106639 a(1)-a(6) are the only members with fewer than 7 prime factors between p-1, p, and p+1. Dickson's conjecture implies that this sequence is infinite. The Bateman-Horn-Stemmler conjecture suggests that there are about 1.905x/(log x)^3 members up to x. - _Charles R Greathouse IV_, Apr 20 2011
%H A106639 T. D. Noe, <a href="/A106639/b106639.txt">Table of n, a(n) for n = 1..1000</a>
%H A106639 L. E. Dickson, <a href="http://oeis.org/wiki/File:A_new_extension_of_Dirichlet%27s_theorem_on_prime_numbers.pdf">A new extension of Dirichlet's theorem on prime numbers</a>, Messenger of Math., 33 (1904), 155-161.
%F A106639 Primes p such that Omega(p^3 - p) <= 7, where Omega is A001222.
%e A106639 19 is in the sequence because 18 has 3 prime factors, 2, 3 and 3;
%e A106639 19 has 1 and 20 has 3 prime factors, 2, 2 and 5, for a total of 7 prime factors in the neighborhood.
%t A106639 Select[Prime[Range[1000]], Total[FactorInteger[#^3 - #]][[2]] <= 7&] (* _T. D. Noe_, Apr 20 2011 *)
%o A106639 (PARI) isA106639(p)=my(g=gcd(p-1,12));isprime(p\g)&isprime((p+1)*g/24)&isprime(p) \\ _Charles R Greathouse IV_, Apr 20 2011
%o A106639 (PARI) forprime(p=1,6000,if(bigomega(p-1)+bigomega(p+1)<=6,print1(p", "))) \\ _Chris Boyd_, Mar 23 2014
%Y A106639 Cf. A239669.
%K A106639 nonn
%O A106639 1,1
%A A106639 _Walter Nissen_, May 11 2005
%E A106639 Formula, comment, offset, program, and link from _Charles R Greathouse IV_, Apr 20 2011