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

1,1

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)<pi(x).

Links

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

Mathematica

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

Prog

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

Crossrefs

Cf. A000040, A001359.

Sequence in context: A050667 A288879 A102906 * A069453 A104071 A288616

Adjacent sequences: A178453 A178454 A178455 * A178457 A178458 A178459

Keyword

nonn

Author

Vladimir Shevelev, Dec 23 2010

Status

approved