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

1,1

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

Mathematica

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

Prog

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

Crossrefs

Cf. A023200, A178456.

Sequence in context: A318295 A165294 A046076 * A144714 A140817 A274518

Adjacent sequences: A178524 A178525 A178526 * A178528 A178529 A178530

Keyword

nonn

Author

Vladimir Shevelev, Dec 23 2010

Status

approved