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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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55
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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;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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Except for the first term, 5, all terms == 11 (mod 30) - Zak Seidov, 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, May 04 2009]
n is the first prime of 2 consecutive twin prime pairs. - Daniel Forgues, Aug 01 2009
The prime quadruples of form p + (0, 2, 6, 8) have the quadruple congruence class (-1, +1, -1, +1) (mod 6). - Daniel Forgues, 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. - Daniel Forgues, Aug 12 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. - Daniel Forgues, Aug 12 2009
Solutions of the equation n'+(n+2)'+(n+6)'+(n+8)'=4, where n' is the arithmetic derivative of n'. [Paolo P. Lava, Nov 09 2012]
Subsequence of A022004. - R. J. Mathar, Feb 10 2013
The quadruplets are listed in A136162. - M. F. Hasler, Apr 20 2013
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REFERENCES
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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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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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
T. R. Nicely, Enumeration to 1.6e15 of the prime quadruplets
Eric Weisstein's World of Mathematics, Prime Quadruplet
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FORMULA
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a(n) = 11 + 30 A014561(n-1) for n>1. [From M. F. Hasler, May 04 2009]
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EXAMPLE
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Contribution 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.
1002341=a(167) is the least quadruplet prime beyond 10^6. (End)
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MAPLE
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A007530:=proc(q)
local n;
for n from 1 to q do
if isprime(n) and isprime (n+2) and isprime(n+6) and isprime (n+8) then print(n); fi;
od; end:
A007530(10000000000); # Paolo P. Lava, Jan 30 2013.
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MATHEMATICA
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A007530 = Select[Range[1, 10^5 - 1, 2], Union[PrimeQ[# + {0, 2, 6, 8}]] == {True} &] (* From Alonso del Arte, Sep 24 2011 *)
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PROG
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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, May 04 2009]
(MAGMA) [ p: p in PrimesUpTo(11000)| IsPrime(p+2) and IsPrime(p+6) and IsPrime(p+8)] [From Vincenzo Librandi, Nov 18 2010]
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CROSSREFS
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Cf. A159910 (first differences divided by 30), A120120.
Cf. A007811, A014561.
Sequence in context: A199325 A199305 A096473 * A157967 A088268 A030085
Adjacent sequences: A007527 A007528 A007529 * 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, Robert G. Wilson v
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EXTENSIONS
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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. [From M. F. Hasler, May 04 2009]
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STATUS
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approved
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