cp's OEIS Frontend

A022007 Initial members of prime 5-tuples (p, p+4, p+6, p+10, p+12).

%I A022007 #72 Feb 16 2025 08:32:34
%S A022007 7,97,1867,3457,5647,15727,16057,19417,43777,79687,88807,101107,
%T A022007 257857,266677,276037,284737,340927,354247,375247,402757,419047,
%U A022007 427237,463447,470077,626617,666427,736357,823717,855727,959467,978067,1022377,1043587,1068247
%N A022007 Initial members of prime 5-tuples (p, p+4, p+6, p+10, p+12).
%C A022007 Subsequence of A052378. - _R. J. Mathar_, Feb 11 2013
%C A022007 All terms are congruent to 7 (modulo 30). - _Matt C. Anderson_, May 22 2015
%C A022007 This sequence is related to the admissible prime 5-tuple (0, 4, 6, 10, 12) because the sequence [1, 2, 3, 1, 2, repeat(1)] gives for n >= 1 the smallest element of RS0(A000040(n)) (the smallest nonnegative complete residue systems modulo prime(n)) which defines a residue class containing none of the 5-tuple members. This 5-tuple is one of two prime constellations of diameter 12. The other one is (0, 2, 6, 8, 12) with initial members given in A022006. See the Wikipedia and Weisstein pages. - _Wolfdieter Lang_, Oct 06 2017
%H A022007 Zak Seidov, <a href="/A022007/b022007.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A022007 T. Forbes and Norman Luhn, <a href="http://www.pzktupel.de/ktuplets.htm">Prime k-tuplets</a>.
%H A022007 Norman Luhn, <a href="http://www.pzktupel.de/smarchive.html">Table of n, a(n) for n = 1..1000000</a>.
%H A022007 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeConstellation.html">Prime Constellation</a>.
%H A022007 Wikipedia, <a href="https://en.wikipedia.org/wiki/Prime_k-tuple">Prime k-tuple</a>.
%F A022007 a(n) = 7 + 30*A089157(n). - _Zak Seidov_, Nov 01 2011
%e A022007 Admissibility guaranteeing sequence [1, 2, 3, 1, 2, repeat(1)] examples: the only residue class modulo prime(3) = 5 which contains none of the 5-tuple (0, 4, 6, 10, 12) members is 3 (mod 5). For prime(5) = 11 the first class is 2 (mod 11); the others are 3, 5, 7, 8, 9 (mod 11). - _Wolfdieter Lang_, Oct 06 2017
%t A022007 Transpose[Select[Partition[Prime[Range[76000]], 5, 1], Differences[#] == {4, 2, 4, 2} &]][[1]] (* _Harvey P. Dale_, Aug 16 2014 *)
%o A022007 (PARI) forprime(p=2,1e5,if(isprime(p+4)&&isprime(p+6)&&isprime(p+10)&&isprime(p+12),print1(p", "))) \\ _Charles R Greathouse IV_, Jul 15 2011
%o A022007 (Magma) [p: p in PrimesUpTo(2*10^6) | IsPrime(p+4) and IsPrime(p+6) and IsPrime(p+10)and IsPrime(p+12)]; // _Vincenzo Librandi_, Aug 23 2015
%o A022007 (Perl) use ntheory ":all"; say for sieve_prime_cluster(1,1e7, 4,6,10,12); # _Dana Jacobsen_, Sep 30 2015
%Y A022007 Cf. A022006, A089157.
%K A022007 nonn,easy
%O A022007 1,1
%A A022007 _Warut Roonguthai_