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

29, 31, 41, 43, 59, 61, 67, 71, 79, 83, 89, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 157, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 389

Offset: 1

Author

Vladimir Shevelev, Dec 23 2010

Keywords

• nonn

Comments

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).

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)

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A001359.

Programs

Mathematica

Select[Prime[Range[100]],PrimeNu[#-1]>2||PrimeNu[#+1]>2&] (* Harvey P. Dale, May 15 2019 *)

PARI

lista(nn) = {forprime(p=2, nn, if ((omega(p-1) > 2) || (omega(p+1) > 2), print1(p, ", ")););} \\ Michel Marcus, Feb 06 2016

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