A098424 -id:A098424 - cp's OEIS frontend

A007529 Prime triples: p; p+2 or p+4; p+6 all prime.

5, 7, 11, 13, 17, 37, 41, 67, 97, 101, 103, 107, 191, 193, 223, 227, 277, 307, 311, 347, 457, 461, 613, 641, 821, 823, 853, 857, 877, 881, 1087, 1091, 1277, 1297, 1301, 1423, 1427, 1447, 1481, 1483, 1487, 1607, 1663, 1693, 1783, 1867, 1871, 1873, 1993, 1997

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

nonn

Or, prime(m) such that prime(m+2) = prime(m)+6. - Zak Seidov, May 07 2012

H. Riesel, "Prime numbers and computer methods for factorization", Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see p. 65.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Zak Seidov, Table of n, a(n) for n = 1..10000
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

Cf. A031924, A023201, A098414, A098415, A098424, A098416, A098417.

Union of A022004 and A022005.

[NthPrime(n): n in [1..310] | (NthPrime(n)+6) eq NthPrime(n+2)]; // Bruno Berselli, May 07 2012

N:= 10000: # to get all terms <= N
Primes:= select(isprime, [seq(2*i+1, i=1..floor((N+5)/2))]):locs:= select(t -> Primes[t+2]-Primes[t]=6, [$1..nops(Primes)-2]):
Primes[locs]; # Robert Israel, Apr 30 2015

ptrsQ[n_]:=PrimeQ[n+6]&&(PrimeQ[n+2]||PrimeQ[n+4])
Select[Prime[Range[400]],ptrsQ]  (* Harvey P. Dale, Mar 08 2011 *)

p=2;q=3;forprime(r=5,1e4,if(r-p==6,print1(p", "));p=q;q=r) \\ Charles R Greathouse IV, May 07 2012

a(n) = A098415(n) - 6. - Zak Seidov, Apr 30 2015

A098415 Greatest members r of prime triples (p,q,r) with p

Original entry on oeis.org

11, 13, 17, 19, 23, 43, 47, 73, 103, 107, 109, 113, 197, 199, 229, 233, 283, 313, 317, 353, 463, 467, 619, 647, 827, 829, 859, 863, 883, 887, 1093, 1097, 1283, 1303, 1307, 1429, 1433, 1453, 1487, 1489, 1493, 1613, 1669, 1699, 1789, 1873, 1877, 1879, 1999

Offset: 1

Reinhard Zumkeller, Sep 07 2004

nonn

Union of A098412 and A098413;

a(n)=A007529(n)+6; either a(n)=A098414(n)+2 or a(n)=A098414(n)+4.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime Triplet

Cf. A098424, A098416, A098417.

Transpose[Select[Partition[Prime[Range[350]],3,1],#[[3]]- #[[1]] == 6&]][[3]] (* Harvey P. Dale, Mar 17 2015 *)

is(n)=isprime(n) && isprime(n-6) && (isprime(n-2) || isprime(n-4)) \\ Charles R Greathouse IV, Feb 23 2017

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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A007529 Prime triples: p; p+2 or p+4; p+6 all prime.

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A098415 Greatest members r of prime triples (p,q,r) with p

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

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