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A178456 Primes p such that p-1 or p+1 has more than two distinct prime divisors.

%I A178456 #28 May 15 2019 11:06:41
%S A178456 29,31,41,43,59,61,67,71,79,83,89,101,103,109,113,127,131,137,139,149,
%T A178456 151,157,167,173,179,181,191,197,199,211,223,227,229,233,239,241,251,
%U A178456 257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,389
%N A178456 Primes p such that p-1 or p+1 has more than two distinct prime divisors.
%C A178456 Sequence contains many pairs of twin primes. More exactly, denote A(x), t(x),T(x) the counting functions of this sequence, twin primes in this sequence and all twin primes correspondingly. In supposition of the infinitude of twin primes, the very plausible conjectures are: (1) for x tends to infinity, t(x)~T(x) and (2) for x >= 31, t(x)/A(x) > T(x)/pi(x).
%C A178456 Indeed (heuristic arguments), the middles of twin pairs (beginning with the second pair) belong to progression {6*n}. Let us choose randomly n. The probability that n has prime divisors 2,3 only is, as well known, O((log n)^2/n), i.e. it is quite natural to conjecture that almost all twin pairs are in the sequence. Besides, it is natural to conjecture that the inequality is true as well, since A(x)<pi(x).
%H A178456 Harvey P. Dale, <a href="/A178456/b178456.txt">Table of n, a(n) for n = 1..10000</a>
%t A178456 Select[Prime[Range[100]],PrimeNu[#-1]>2||PrimeNu[#+1]>2&] (* _Harvey P. Dale_, May 15 2019 *)
%o A178456 (PARI) lista(nn) = {forprime(p=2, nn, if ((omega(p-1) > 2) || (omega(p+1) > 2), print1(p, ", ")););} \\ _Michel Marcus_, Feb 06 2016
%Y A178456 Cf. A000040, A001359.
%K A178456 nonn
%O A178456 1,1
%A A178456 _Vladimir Shevelev_, Dec 23 2010