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

%I #33 Feb 09 2025 06:35:12

%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