OFFSET
1,1
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
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.
FORMULA
Primes p such that Omega(p^3 - p) <= 7, where Omega is A001222.
EXAMPLE
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.
MATHEMATICA
Select[Prime[Range[1000]], Total[FactorInteger[#^3 - #]][[2]] <= 7&] (* T. D. Noe, Apr 20 2011 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Walter Nissen, May 11 2005
EXTENSIONS
Formula, comment, offset, program, and link from Charles R Greathouse IV, Apr 20 2011
STATUS
approved