A023200 Primes p such that p + 4 is also prime.

3, 7, 13, 19, 37, 43, 67, 79, 97, 103, 109, 127, 163, 193, 223, 229, 277, 307, 313, 349, 379, 397, 439, 457, 463, 487, 499, 613, 643, 673, 739, 757, 769, 823, 853, 859, 877, 883, 907, 937, 967, 1009, 1087, 1093, 1213, 1279, 1297, 1303, 1423, 1429, 1447, 1483

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

Smaller member p of cousin prime pairs (p, p+4).

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

Apart from the first term, all terms are of the form 6n + 1.

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

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

From Alonso del Arte, Jan 12 2019: (Start)

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.

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.

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.

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)

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Andrew Granville and Greg Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
• Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
• Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
• H. J. Weber, A Sieve for Cousin Primes, arXiv:1204.3795v1 [math.NT], 2012.
• Eric Weisstein's World of Mathematics, Cousin Primes
• Eric Weisstein's World of Mathematics, Twin Primes
• Wikipedia, Cousin prime.
• Index entries for primes, gaps between

Crossrefs

Exactly the same as A029710 except for the exclusion of 3.

Cf. A000010, A003557, A007947, A046132, A098429, A000040, A010051.

Programs

Haskell

a023200 n = a023200_list !! (n-1)
a023200_list = filter ((== 1) . a010051') $
               map (subtract 4) $ drop 2 a000040_list
-- Reinhard Zumkeller, Aug 01 2014

Magma

[p: p in PrimesUpTo(1500) | NextPrime(p)-p eq 4]; // Bruno Berselli, Apr 09 2013

Maple

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

Mathematica

Select[Range[10^2], PrimeQ[#] && PrimeQ[# + 4] &] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Select[Prime[Range[250]],PrimeQ[#+4]&] (* Harvey P. Dale, Oct 09 2023 *)

PARI

print1(3);p=7;forprime(q=11,1e3,if(q-p==4,print1(", "p)); p=q) \\ Charles R Greathouse IV, Mar 20 2013

Formula

a(n) = A046132(n) - 4 = A087679(n) - 2.

a(n) >> n log^2 n via the Selberg sieve. - Charles R Greathouse IV, Nov 20 2016

Extensions

Definition modified by Vincenzo Librandi, Aug 02 2009

Definition revised by N. J. A. Sloane, Mar 05 2010