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%I #82 Oct 09 2023 15:13:53
%S 3,7,13,19,37,43,67,79,97,103,109,127,163,193,223,229,277,307,313,349,
%T 379,397,439,457,463,487,499,613,643,673,739,757,769,823,853,859,877,
%U 883,907,937,967,1009,1087,1093,1213,1279,1297,1303,1423,1429,1447,1483
%N Primes p such that p + 4 is also prime.
%C Smaller member p of cousin prime pairs (p, p+4).
%C A015913 contains the composite number 305635357, so it is different from both the present sequence and A029710. (305635357 is the only composite member of A015913 < 10^9.) - _Jud McCranie_, Jan 07 2001
%C Apart from the first term, all terms are of the form 6n + 1.
%C Complement of A067775 (primes p such that p + 4 is composite) with respect to A000040 (primes). With prime 2 also primes p such that q^2 + p is prime for some prime q (q = 3 if p = 2, q = 2 if p > 2). Subsequence of A232012. - _Jaroslav Krizek_, Nov 23 2013
%C Conjecture: The sequence is infinite and for every n, a(n+1) < a(n)^(1+1/n). Namely a(n)^(1/n) is a strictly decreasing function of n. - _Jahangeer Kholdi_ and _Farideh Firoozbakht_, Nov 24 2014
%C From _Alonso del Arte_, Jan 12 2019: (Start)
%C If p splits in Z[sqrt(-2)], p + 4 is an inert prime in that domain. Likewise, if p splits in Z[sqrt(2)], p + 4 is an inert prime in that domain.
%C The only way for p or p + 4 to split in both domains is if it is congruent to 1 modulo 24, in which case the other prime is inert in both domains.
%C For example, 3 = (1 - sqrt(-2))*(1 + sqrt(-2)) but is inert in Z[sqrt(2)], while 7 = (3 - sqrt(2))*(3 + sqrt(2)) but is inert in Z[sqrt(-2)]. And also 11 = (3 - sqrt(-2))*(3 + sqrt(-2)) but 15 is composite in Z or any quadratic integer ring.
%C And 97 = (5 - 6*sqrt(-2))*(5 + 6*sqrt(-2)) = (1 - 7*sqrt(2))*(1 + 7*sqrt(2)), but 101 is inert in both Z[sqrt(-2)] and Z[sqrt(2)]. (End)
%H T. D. Noe, <a href="/A023200/b023200.txt">Table of n, a(n) for n = 1..10000</a>
%H Andrew Granville and Greg Martin, <a href="https://arxiv.org/abs/math/0408319">Prime number races</a>, arXiv:math/0408319 [math.NT], 2004.
%H Ernest G. Hibbs, <a href="https://www.proquest.com/openview/4012f0286b785cd732c78eb0fc6fce80">Component Interactions of the Prime Numbers</a>, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
%H Maxie D. Schmidt, <a href="https://arxiv.org/abs/1701.04741">New Congruences and Finite Difference Equations for Generalized Factorial Functions</a>, arXiv:1701.04741 [math.CO], 2017.
%H H. J. Weber, <a href="http://arxiv.org/abs/1204.3795">A Sieve for Cousin Primes</a>, arXiv:1204.3795v1 [math.NT], 2012.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CousinPrimes.html">Cousin Primes</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TwinPrimes.html">Twin Primes</a>
%H <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>
%F a(n) = A046132(n) - 4 = A087679(n) - 2.
%F a(n) >> n log^2 n via the Selberg sieve. - _Charles R Greathouse IV_, Nov 20 2016
%p A023200 := proc(n) option remember; if n = 1 then 3; else p := nextprime(procname(n-1)) ; while not isprime(p+4) do p := nextprime(p) ; end do: p ; end if; end proc: # _R. J. Mathar_, Sep 03 2011
%t Select[Range[10^2], PrimeQ[#] && PrimeQ[# + 4] &] (* _Vladimir Joseph Stephan Orlovsky_, Apr 29 2008 *)
%t Select[Prime[Range[250]],PrimeQ[#+4]&] (* _Harvey P. Dale_, Oct 09 2023 *)
%o (PARI) print1(3);p=7;forprime(q=11,1e3,if(q-p==4,print1(", "p)); p=q) \\ _Charles R Greathouse IV_, Mar 20 2013
%o (Magma) [p: p in PrimesUpTo(1500) | NextPrime(p)-p eq 4]; // _Bruno Berselli_, Apr 09 2013
%o (Haskell)
%o a023200 n = a023200_list !! (n-1)
%o a023200_list = filter ((== 1) . a010051') $
%o map (subtract 4) $ drop 2 a000040_list
%o -- _Reinhard Zumkeller_, Aug 01 2014
%Y Exactly the same as A029710 except for the exclusion of 3.
%Y Cf. A000010, A003557, A007947, A046132, A098429, A000040, A010051.
%K nonn
%O 1,1
%A _David W. Wilson_
%E Definition modified by _Vincenzo Librandi_, Aug 02 2009
%E Definition revised by _N. J. A. Sloane_, Mar 05 2010
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