A106639 - OEIS

%I #23 Mar 05 2015 14:38:57

%S 2,3,5,7,11,13,19,23,29,37,43,59,61,67,83,157,173,227,277,283,317,347,

%T 563,653,733,787,877,907,997,1213,1237,1283,1307,1523,1867,2083,2693,

%U 2797,2803,3253,3413,3517,3643,3677,3733,3803,4253,4363,4547,4723,5387

%N Distinguished primes.

%C 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 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 T. D. Noe, <a href="/A106639/b106639.txt">Table of n, a(n) for n = 1..1000</a>

%H 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 Primes p such that Omega(p^3 - p) <= 7, where Omega is A001222.

%e 19 is in the sequence because 18 has 3 prime factors, 2, 3 and 3;

%e 19 has 1 and 20 has 3 prime factors, 2, 2 and 5, for a total of 7 prime factors in the neighborhood.

%t Select[Prime[Range[1000]], Total[FactorInteger[#^3 - #]][[2]] <= 7&] (* _T. D. Noe_, Apr 20 2011 *)

%o (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 (PARI) forprime(p=1,6000,if(bigomega(p-1)+bigomega(p+1)<=6,print1(p", "))) \\ _Chris Boyd_, Mar 23 2014

%Y Cf. A239669.

%K nonn

%O 1,1

%A _Walter Nissen_, May 11 2005

%E Formula, comment, offset, program, and link from _Charles R Greathouse IV_, Apr 20 2011