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%I #48 Sep 08 2022 08:44:58
%S 7,13,19,23,31,37,43,47,53,61,67,73,79,83,89,97,103,109,113,127,131,
%T 139,151,157,163,167,173,181,193,199,211,223,229,233,241,251,257,263,
%U 271,277,283,293,307,313,317,331,337,349,353,359,367,373,379,383,389
%N Odd primes p such that p+2 is composite.
%C Primes p such that nextprime(p)-p >= 4.
%C Primes p such that p+2 divides (p-1)!.
%C Odd primes n such that n!*B(n+1) is an integer, where B(k) are the Bernoulli numbers. - _Benoit Cloitre_, Feb 06 2002
%C 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
%C 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
%C 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
%C Numbers k such that 4((k-1)! + 1) == k^2 (mod k(k+2)). - _Thomas Ordowski_, May 09 2020
%H T. D. Noe, <a href="/A049591/b049591.txt">Table of n, a(n) for n = 1..1000</a>
%H K. Soundararajan, <a href="http://dx.doi.org/10.1090/S0273-0979-06-01142-6">Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim</a>, Bull. Amer. Math. Soc., 44 (2007), 1-18.
%H <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>
%e 13 is here because it is prime and 15 is composite. Also 15 divides 12!.
%p 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;
%t Select[Prime[Range[100]], NextPrime[#] -#>=4 &] (* _G. C. Greubel_, Aug 22 2019 *)
%o (PARI) isok(p) = isprime(p) && (p % 2) && !isprime(p+2); \\ _Michel Marcus_, Feb 25 2014
%o (Magma) [k:k in PrimesInInterval(3,400)| not IsPrime(k+2)]; // _Marius A. Burtea_, Aug 03 2019
%o (Sage) [nth_prime(n) for n in (1..100) if (nth_prime(n+1) - nth_prime(n)) >= 4] # _G. C. Greubel_, Aug 22 2019
%Y Cf. A067774, A105399.
%K nonn
%O 1,1
%A _Labos Elemer_
%E More terms from _Benoit Cloitre_, Jun 04 2003
%E Edited by _Don Reble_, Dec 20 2006