A098429 -id:A098429 - cp's OEIS frontend

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

David W. Wilson

nonn

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)

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

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

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

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

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

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

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 *)

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

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

Definition modified by Vincenzo Librandi, Aug 02 2009

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

A046132 Larger member p+4 of cousin primes (p, p+4).

Original entry on oeis.org

7, 11, 17, 23, 41, 47, 71, 83, 101, 107, 113, 131, 167, 197, 227, 233, 281, 311, 317, 353, 383, 401, 443, 461, 467, 491, 503, 617, 647, 677, 743, 761, 773, 827, 857, 863, 881, 887, 911, 941, 971, 1013, 1091, 1097, 1217, 1283, 1301, 1307, 1427, 1433

Offset: 1

Eric W. Weisstein

nonn

A pair of cousin primes are primes of the form p and p+4 (where p+2 may or may not be a prime). - N. J. A. Sloane, Mar 18 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Cousin Primes
Wikipedia, Cousin prime.

Essentially the same as A031505. Cf. A023200, A029710, A098429.

Cf. A000040, A010051.

a046132 n = a046132_list !! (n-1)
a046132_list = filter ((== 1) . a010051') $ map (+ 4) a000040_list
-- Reinhard Zumkeller, Aug 01 2014

lst={};Do[p=Prime[n];If[PrimeQ[p4=p+4], (*Print[p4];*)AppendTo[lst, p4]], {n, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)
Select[Prime[Range[300]],PrimeQ[#+4]&]+4 (* Harvey P. Dale, Dec 15 2017 *)

forprime(p=2,1e5,if(isprime(p-4),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = A023200(n) + 4 = A087679(n) + 2.

a(n) = 3*A157834(n-1) + 2 = A029710(n-1) + 4 = 6*A056956(n-1) + 5 (thus a(n) mod 6 == 5), for all n>1. - M. F. Hasler, Jan 15 2013

A098428 Number of sexy prime pairs (p, p+6) with p <= n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15

Offset: 1

Reinhard Zumkeller, Sep 07 2004

nonn

Since there are 2 congruence classes of sexy prime pairs, (-1, -1) (mod 6) and (+1, +1) (mod 6), the number of sexy prime pairs up to n is the sum of the number of sexy prime pairs for each class, expected to be asymptotically the same for both (with the expected Chebyshev bias against the quadratic residue class (+1, +1) (mod 6), which doesn't affect the asymptotic distribution among the 2 classes). - Daniel Forgues, Aug 05 2009

			The first sexy prime pairs are: (5,11), (7,13), (11,17), (13,19), ...
therefore the sequence starts: 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 4, ...

Daniel Forgues, Table of n, a(n) for n=1..99994
Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021]

Cf. A023201, A046117, A098424, A071538, A098429.

Accumulate[Table[If[PrimeQ[n]&&PrimeQ[n+6],1,0],{n,100}]] (* Harvey P. Dale, Feb 08 2015 *)

apply( {A098428(n,o=2,q=o,c)=forprime(p=1+q, n+6, (o+6==p)+((o=q)+6==q=p) && c++);c}, [1..99]) \\ M. F. Hasler, Jan 02 2020
[#[p:p in PrimesInInterval(1,n)| IsPrime(p+6)]:n in [1..100]]; // Marius A. Burtea, Jan 03 2020

a(n) = # { p in A023201 | p <= n } = number of elements in intersection of A023201 and [1,n]. - M. F. Hasler, Jan 02 2020

Edited by Daniel Forgues, Aug 01 2009, M. F. Hasler, Jan 02 2020

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

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A046132 Larger member p+4 of cousin primes (p, p+4).

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A098428 Number of sexy prime pairs (p, p+6) with p <= n.

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