cp's OEIS Frontend

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

%I A178527 #33 Feb 09 2025 06:35:12
%S A178527 103,107,163,167,193,197,229,233,257,271,283,313,317,347,359,383,397,
%T A178527 401,431,433,457,463,467,523,557,563,587,593,607,613,617,643,647,653,
%U A178527 661,691,733,739,743,757,761,797,821,823,827
%N A178527 Primes p such that either p - 2 or p + 2 has more than two distinct prime divisors.
%C A178527 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 A178527 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 A178527 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 A178527 Probably this sequence contains almost all primes and so a(n) ~ n log n. - _Charles R Greathouse IV_, Sep 24 2013
%H A178527 Charles R Greathouse IV, <a href="/A178527/b178527.txt">Table of n, a(n) for n = 1..10000</a>
%t A178527 Select[Prime[Range[200]], PrimeNu[# - 2] > 2 || PrimeNu[# + 2] > 2 &] (* _Alonso del Arte_, Dec 23 2010 *)
%o A178527 (PARI) is(n)=isprime(n) && n>9 && (omega(n-2)>2||omega(n+2)>2) \\ _Charles R Greathouse IV_, Sep 24 2013
%Y A178527 Cf. A023200, A178456.
%K A178527 nonn
%O A178527 1,1
%A A178527 _Vladimir Shevelev_, Dec 23 2010