A178527 - OEIS

%I #30 Aug 20 2017 23:18:13

%S 103,107,163,167,193,197,229,233,257,271,283,313,317,347,359,383,397,

%T 401,431,433,457,463,467,523,557,563,587,593,607,613,617,643,647,653,

%U 661,691,733,739,743,757,761,797,821,823,827

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

%C 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.

%C 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.

%C 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).

%C Probably this sequence contains almost all primes and so a(n) ~ n log n. - _Charles R Greathouse IV_, Sep 24 2013

%H Charles R Greathouse IV, <a href="/A178527/b178527.txt">Table of n, a(n) for n = 1..10000</a>

%t Select[Prime[Range[200]], PrimeNu[# - 2] > 2 || PrimeNu[# + 2] > 2 &] (* _Alonso del Arte_, Dec 23 2010 *)

%o (PARI) is(n)=isprime(n) && n>9 && (omega(n-2)>2||omega(n+2)>2) \\ _Charles R Greathouse IV_, Sep 24 2013

%Y Cf. A023200, A178456.

%K nonn

%O 1,1

%A _Vladimir Shevelev_, Dec 23 2010