A007530 Prime quadruples: numbers k such that k, k+2, k+6, k+8 are all prime.
5, 11, 101, 191, 821, 1481, 1871, 2081, 3251, 3461, 5651, 9431, 13001, 15641, 15731, 16061, 18041, 18911, 19421, 21011, 22271, 25301, 31721, 34841, 43781, 51341, 55331, 62981, 67211, 69491, 72221, 77261, 79691, 81041, 82721, 88811, 97841, 99131
Offset: 1
Keywords
Examples
From _M. F. Hasler_, May 04 2009: (Start)
a(1)=5 is the start of the first prime quadruplet, {5,7,11,13}.
a(2)=11 is the start of the second prime quadruplet, {11,13,17,19}, and all other prime quadruplets differ from this one by a multiple of 30.
a(100)=470081 is the start of the 100th prime quadruplet;
a(500)=4370081 is the start of the 500th prime quadruplet.
a(167)=1002341 is the least quadruplet prime beyond 10^6. (End)
References
- H. Rademacher, Lectures on Elementary Number Theory. Blaisdell, NY, 1964, p. 4.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Matt C. Anderson, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe).
- C. K. Caldwell, The Prime Glossary, prime quadruple
- Tony Forbes and Norman Luhn, prime k-tuplets
- Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
- Norman Luhn, Table of n, a(n) for n = 1..1000000
- Thomas R. Nicely, Enumeration to 1.6e15 of the prime quadruplets
- H. Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics, Vol. 57, Birkhäuser, Boston, 1985, ISBN: 978-0-8176-8297-2, Chap. 4, see p. 65.
- Eric Weisstein's World of Mathematics, Prime Quadruplet
Programs
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Magma
[ p: p in PrimesUpTo(11000)| IsPrime(p+2) and IsPrime(p+6) and IsPrime(p+8)] // Vincenzo Librandi, Nov 18 2010
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Mathematica
A007530 = Select[Range[1, 10^5 - 1, 2], Union[PrimeQ[# + {0, 2, 6, 8}]] == {True} &] (* Alonso del Arte, Sep 24 2011 *) Select[Prime[Range[10000]],AllTrue[#+{2,6,8},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 11 2019 *)
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PARI
A007530( n, print_all=0, s=2 )={ my(p,q,r); until(!n--, until( p+8==s=nextprime(s+2), p=q; q=r; r=s); print_all && print1(p","));p} \\ The optional 3rd argument can be used to obtain large values by starting from some precomputed point instead of zero, using a(n+k) = A007530(k+1,,a(n)) (or A007530(k,,a(n)-1) for k>0); e.g., you get a(10^4+k) using A007530(k+1,,265201421) (value of a(10^4) from the comments section). - M. F. Hasler, May 04 2009
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PARI
forprime(p=2, 10^5, if(isprime(p+2) && isprime(p+6) && isprime(p+8), print1(p, ", "))) \\ Felix Fröhlich, Jun 22 2014
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Python
from sympy import primerange def aupto(limit): p, q, r, alst = 2, 3, 5, [] for s in primerange(7, limit+9): if p+2 == q and p+6 == r and p+8 == s: alst.append(p) p, q, r = q, r, s return alst print(aupto(10**5)) # Michael S. Branicky, May 11 2021
Formula
a(n) = 11 + 30*A014561(n-1) for n > 1. - M. F. Hasler, May 04 2009
Extensions
More terms from Warut Roonguthai
Incorrect formula and Mathematica program removed by N. J. A. Sloane, Dec 04 2008, at the suggestion of Zak Seidov
Values up to a(1000) checked with the given PARI code by M. F. Hasler, May 04 2009
Comments