A049591 Odd primes p such that p+2 is composite.

7, 13, 19, 23, 31, 37, 43, 47, 53, 61, 67, 73, 79, 83, 89, 97, 103, 109, 113, 127, 131, 139, 151, 157, 163, 167, 173, 181, 193, 199, 211, 223, 229, 233, 241, 251, 257, 263, 271, 277, 283, 293, 307, 313, 317, 331, 337, 349, 353, 359, 367, 373, 379, 383, 389

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Comments

Primes p such that nextprime(p)-p >= 4.

Primes p such that p+2 divides (p-1)!.

Odd primes n such that n!*B(n+1) is an integer, where B(k) are the Bernoulli numbers. - Benoit Cloitre, Feb 06 2002

Sequence appears also to give all n > 1 such that there is no prime p satisfying the inequality n < p < n+tau(n)^2 where tau(n)=A000005(n). - Benoit Cloitre, Apr 13 2002

Conjecture: start from any initial value f(1) >= 2 and define f(n) to be the largest prime factor of f(1) +f(2) + ... +f(n-1); then f(n) = n/2 + O(log(n)) and there are infinitely many primes p such that f(2p)=p. Conjecture: current sequence gives primes satisfying f(2p)=p when f(1)=3. - Benoit Cloitre, Jun 04 2003

Numbers k such that 2((k-1)! + 1) + k is divisible by k(k+2). For 7 and 13, the respective quotients are also in the sequence. Are there any other such k? - Ivan N. Ianakiev, Aug 03 2019. The next values of k with respective quotients in this sequence are 103, 1531, 1637. - Amiram Eldar, Jun 08 2020

Numbers k such that 4((k-1)! + 1) == k^2 (mod k(k+2)). - Thomas Ordowski, May 09 2020

Examples

			13 is here because it is prime and 15 is composite. Also 15 divides 12!.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc., 44 (2007), 1-18.
• Index entries for primes, gaps between

Crossrefs

Cf. A067774, A105399.

Programs

Magma

[k:k in PrimesInInterval(3,400)| not IsPrime(k+2)]; // Marius A. Burtea, Aug 03 2019

Maple

d:=4; M:=1000; t0:=[]; for n from 1 to M do p:=ithprime(n); if nextprime(p) - p >= d then t0:=[op(t0),p]; fi; od: t0;

Mathematica

Select[Prime[Range[100]], NextPrime[#] -#>=4 &] (* G. C. Greubel, Aug 22 2019 *)

PARI

isok(p) = isprime(p) && (p % 2) && !isprime(p+2); \\ Michel Marcus, Feb 25 2014

Sage

[nth_prime(n) for n in (1..100) if (nth_prime(n+1) - nth_prime(n)) >= 4] # G. C. Greubel, Aug 22 2019

Extensions

More terms from Benoit Cloitre, Jun 04 2003

Edited by Don Reble, Dec 20 2006