A178527 Primes p such that either p - 2 or p + 2 has more than two distinct prime divisors.

103, 107, 163, 167, 193, 197, 229, 233, 257, 271, 283, 313, 317, 347, 359, 383, 397, 401, 431, 433, 457, 463, 467, 523, 557, 563, 587, 593, 607, 613, 617, 643, 647, 653, 661, 691, 733, 739, 743, 757, 761, 797, 821, 823, 827

Offset: 1

Author

Vladimir Shevelev, Dec 23 2010

Keywords

• nonn

Comments

Sequence contains "many" pairs of cousin primes. More exactly, our conjectures are: (1) sequence contains almost all cousin primes; (2)for x >= 107, c(x)/A(x) > C(x)/pi(x), where A(x), c(x) and C(x) are the counting functions for this sequence, cousin pairs in this sequence and all cousin pairs respectively.

Indeed (a heuristic argument), a number n in the middle of a randomly chosen pair of cousin primes may be considered as a random integer.

The probability that n has no more than two prime divisors is, as well known, O(log(log(n))/log(n)), i.e., it is natural to conjecture that almost all cousin pairs are in the sequence. Furthermore, it is natural to conjecture that the inequality is true as well, since A(x) < pi(x).

Probably this sequence contains almost all primes and so a(n) ~ n log n. - Charles R Greathouse IV, Sep 24 2013

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A023200, A178456.

Programs

Mathematica

Select[Prime[Range[200]], PrimeNu[# - 2] > 2 || PrimeNu[# + 2] > 2 &] (* Alonso del Arte, Dec 23 2010 *)

PARI

is(n)=isprime(n) && n>9 && (omega(n-2)>2||omega(n+2)>2) \\ Charles R Greathouse IV, Sep 24 2013