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Search: id:A007530
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| A007530 |
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Prime quadruples: numbers n such that n, n+2, n+6, n+8 are all prime.
(Formerly M3816)
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+0
44
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| 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
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Except for the first term, 5, all terms == 11 (mod 30) - Zak Seidov (zakseidov(AT)yahoo.com), Dec 04 2008
Some further values: For k=1,...,10, a(k*10^3)=11721791, 31210841, 54112601, 78984791, 106583831, 136466501, 165939791, 196512551, 230794301, 265201421. [From M. F. Hasler (MHasler(AT)univ-ag.fr), May 04 2009]
n is the first prime of 2 consecutive twin prime pairs. [From Daniel Forgues (squid(AT zensearch.com), Aug 01 2009]
The prime quadruples of form p + (0, 2, 6, 8) have the quadruple congruence class (-1, +1, -1, +1) (mod 6). [From Daniel Forgues (squid(AT zensearch.com), Aug 12 2009]
s = (p+8)-(p) = 8 is the smallest s giving an admissible prime quadruple form, for which the only admissible form is p + (0, 2, 6, 8), since (0, 2, 6, 8) is the only form not covering all the congruence classes for any prime <= 4. Since s is smallest, these prime quadruples are prime constellations (or prime quadruplets), i.e. they contain consecutive primes. [From Daniel Forgues (squid(AT zensearch.com), Aug 012 2009]
Except for the first term, 5, all prime quadruples are of the form (15k-4, 15k-2, 15k+2, 15k+4), with k >= 1, and so are centered on 15k. [From Daniel Forgues (squid(AT zensearch.com), Aug 012 2009]
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. Rademacher, Lectures on Elementary Number Theory. Blaisdell, NY, 1964, p. 4.
H. Riesel, ``Prime numbers and computer methods for factorization,'' Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see p. 65.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
C. K. Caldwell, The Prime Glossary, prime quadruple
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
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a(n) = 11 + 30 A014561(n-1) for n>1. [From M. F. Hasler (MHasler(AT)univ-ag.fr), May 04 2009]
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EXAMPLE
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Contribution from M. F. Hasler (MHasler(AT)univ-ag.fr), 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.
1002341=a(167) is the least quadruplet prime beyond 10^6. (End)
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PROGRAM
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(PARI) A007530( n, list=0, s=2 )={ my(p, q, r); until(!n--, until( p+8==s=nextprime(s+2), p=q; q=r; r=s); list & print1(p", ")); p} /* NB: a(n+k)=A007530(k+1, , a(n)) (=A007530(k, a(n)-1) for k>0), e.g. A007530(k+1, 265201421)=a(10^4+k). */ [From M. F. Hasler (MHasler(AT)univ-ag.fr), May 04 2009]
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CROSSREFS
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Cf. A159910 (first differences divided by 30). [From M. F. Hasler (MHasler(AT)univ-ag.fr), May 04 2009]
Sequence in context: A030079 A066596 A096473 this_sequence A157967 A088268 A030085
Adjacent sequences: A007527 A007528 A007529 this_sequence A007531 A007532 A007533
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com)
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EXTENSIONS
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More terms from Warut Roonguthai (warut822(AT)yahoo.com)
Incorrect formula and Mathematica program removed by N. J. A. Sloane (njas(AT)research.att.com), Dec 04 2008, at the suggestion of Zak Seidov
Values up to a(1000) checked with the given PARI code. [From M. F. Hasler (MHasler(AT)univ-ag.fr), May 04 2009]
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