A106639 Distinguished primes.

2, 3, 5, 7, 11, 13, 19, 23, 29, 37, 43, 59, 61, 67, 83, 157, 173, 227, 277, 283, 317, 347, 563, 653, 733, 787, 877, 907, 997, 1213, 1237, 1283, 1307, 1523, 1867, 2083, 2693, 2797, 2803, 3253, 3413, 3517, 3643, 3677, 3733, 3803, 4253, 4363, 4547, 4723, 5387

Offset: 1

Author

Walter Nissen, May 11 2005

Keywords

• nonn

Comments

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.

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

Examples

			19 is in the sequence because 18 has 3 prime factors, 2, 3 and 3;
19 has 1 and 20 has 3 prime factors, 2, 2 and 5, for a total of 7 prime factors in the neighborhood.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• L. E. Dickson, A new extension of Dirichlet's theorem on prime numbers, Messenger of Math., 33 (1904), 155-161.

Crossrefs

Cf. A239669.

Programs

Mathematica

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

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

PARI

forprime(p=1,6000,if(bigomega(p-1)+bigomega(p+1)<=6,print1(p", "))) \\ Chris Boyd, Mar 23 2014

Formula

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

Extensions

Formula, comment, offset, program, and link from Charles R Greathouse IV, Apr 20 2011