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A106639 Distinguished primes. 3
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 (list; graph; refs; listen; history; text; internal format)
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

Cf. A239669.

Sequence in context: A005728 A050437 A096246 * A233462 A233893 A232824

Adjacent sequences:  A106636 A106637 A106638 * A106640 A106641 A106642

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

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Last modified April 2 19:05 EDT 2020. Contains 333190 sequences. (Running on oeis4.)