Search: id:A007530
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%I A007530 M3816
%S A007530 5,11,101,191,821,1481,1871,2081,3251,3461,5651,9431,13001,15641,
%T A007530 15731,16061,18041,18911,19421,21011,22271,25301,31721,34841,43781,
%U A007530 51341,55331,62981,67211,69491,72221,77261,79691,81041,82721,88811,97841,
               99131
%N A007530 Prime quadruples: numbers n such that n, n+2, n+6, n+8 are all prime.
%C A007530 Except for the first term, 5, all terms == 11 (mod 30) - Zak Seidov (zakseidov(AT)yahoo.com), 
               Dec 04 2008
%C A007530 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]
%C A007530 n is the first prime of 2 consecutive twin prime pairs. [From Daniel 
               Forgues (squid(AT zensearch.com), Aug 01 2009]
%C A007530 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]
%C A007530 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]
%C A007530 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]
%D A007530 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007530 H. Rademacher, Lectures on Elementary Number Theory. Blaisdell, NY, 1964, 
               p. 4.
%D A007530 H. Riesel, ``Prime numbers and computer methods for factorization,'' 
               Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 
               4, see p. 65.
%H A007530 T. D. Noe, <a href="b007530.txt">Table of n, a(n) for n=1..1000</a>
%H A007530 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/
               page.php?sort=Quadruple">prime quadruple</a>
%H A007530 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PrimeQuadruplet.html">Link to a section of The World of Mathematics.</
               a>
%F A007530 a(n) = 11 + 30 A014561(n-1) for n>1. [From M. F. Hasler (MHasler(AT)univ-ag.fr), 
               May 04 2009]
%e A007530 Contribution from M. F. Hasler (MHasler(AT)univ-ag.fr), May 04 2009: 
               (Start)
%e A007530 a(1)=5 is the start of the first prime quadruplet, {5,7,11,13}.
%e A007530 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.
%e A007530 a(100)=470081 is the start of the 100th prime quadruplet;
%e A007530 a(500)=4370081 is the start of the 500th prime quadruplet.
%e A007530 1002341=a(167) is the least quadruplet prime beyond 10^6. (End)
%o A007530 (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]
%Y A007530 Cf. A159910 (first differences divided by 30). [From M. F. Hasler (MHasler(AT)univ-ag.fr), 
               May 04 2009]
%Y A007530 Sequence in context: A030079 A066596 A096473 this_sequence A157967 A088268 
               A030085
%Y A007530 Adjacent sequences: A007527 A007528 A007529 this_sequence A007531 A007532 
               A007533
%K A007530 nonn
%O A007530 1,1
%A A007530 N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com)
%E A007530 More terms from Warut Roonguthai (warut822(AT)yahoo.com)
%E A007530 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
%E A007530 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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