A178456 -id:A178456 - cp's OEIS frontend

A178527 Primes p such that either p - 2 or p + 2 has more than two distinct prime divisors.

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

Offset: 1

Vladimir Shevelev, Dec 23 2010

nonn

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

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

Cf. A023200, A178456.

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

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

A178668 Maximal prime divisor of the average of the twin prime pairs, different from 2 and 3. In case of maximal prime divisor is 2 or 3, then a(n)=1.

Original entry on oeis.org

1, 1, 1, 1, 5, 7, 5, 1, 17, 1, 23, 5, 5, 1, 11, 19, 5, 5, 47, 13, 29, 7, 1, 11, 29, 19, 5, 103, 107, 11, 5, 137, 23, 13, 7, 17, 43, 7, 59, 13, 1, 41, 71, 43, 31, 11, 17, 11, 19, 31, 67, 5, 139, 283, 41, 149, 13, 313, 23, 13, 37, 13, 347, 29, 11, 71, 17, 373, 7, 11, 13, 397, 17, 1, 443, 7, 113, 13, 31, 467, 11, 5, 13, 11, 271, 181, 11, 37, 7, 281, 113, 577, 17, 7, 59, 593, 199, 17, 157, 13

Offset: 1

Vladimir Shevelev, Dec 25 2010

nonn

78 from the first 100 terms are first or second members of twin pairs and only 12 are not. In a natural supposition that for large prime terms the latter should be in the majority, there are reasons to assume that the number N for which it occurs for the first time is very large.

The average of a twin-prime pair is the same as 1 + the lower twin prime, whose largest prime factor is tabulated in A060210.

Cf. A178456, A178527, A001359

s = Plus @@@ Select[ Partition[ Prime@ Range@ 350, 2, 1], #[[1]] + 2 == #[[2]] &]; f[n_] := Max[First /@ FactorInteger@ n] /. {2 -> 1, 3 -> 1}, f /@ s

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A178527 Primes p such that either p - 2 or p + 2 has more than two distinct prime divisors.

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