

A178527


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


2



103, 107, 163, 167, 193, 197, 229, 233, 257, 271, 283, 313, 317, 347, 359, 383, 397, 401, 431, 433, 457, 463, 467, 523, 557, 563, 587, 593, 607, 613, 617, 643, 647, 653, 661, 691, 733, 739, 743, 757, 761, 797, 821, 823, 827
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OFFSET

1,1


COMMENTS

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.
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.
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).
Probably this sequence contains almost all primes and so a(n) ~ n log n.  Charles R Greathouse IV, Sep 24 2013


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000


MATHEMATICA

Select[Prime[Range[200]], PrimeNu[#  2] > 2  PrimeNu[# + 2] > 2 &] (* Alonso del Arte, Dec 23 2010 *)


PROG

(PARI) is(n)=isprime(n) && n>9 && (omega(n2)>2omega(n+2)>2) \\ Charles R Greathouse IV, Sep 24 2013


CROSSREFS

Cf. A023200, A178456.
Sequence in context: A318295 A165294 A046076 * A144714 A140817 A274518
Adjacent sequences: A178524 A178525 A178526 * A178528 A178529 A178530


KEYWORD

nonn


AUTHOR

Vladimir Shevelev, Dec 23 2010


STATUS

approved



