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A172456 Primes p such that (p, p+2, p+6, p+12, p+14, p+20) is a prime sextuple.

%I A172456 #36 Feb 16 2025 08:33:11
%S A172456 17,1277,1607,3527,4637,71327,97367,113147,191447,290657,312197,
%T A172456 416387,418337,421697,450797,566537,795647,886967,922067,1090877,
%U A172456 1179317,1300127,1464257,1632467,1749257,1866857,1901357,2073347,2322107
%N A172456 Primes p such that (p, p+2, p+6, p+12, p+14, p+20) is a prime sextuple.
%C A172456 The last digit of each of these prime numbers is 7.
%C A172456 Subsequence of A078946.
%C A172456 The primes are always consecutive: The few ways of inserting other primes are: (p,p+2,p+4)... [impossible since one of these would be a multiple of 3]; (p,p+2,p+6),(p+8),(p+12),(p+14) [impossible since one of these would be a multiple of 5]; (p,p+2,p+6),(p+10) [impossible since one of these would be a multiple of 3]; (p,p+2,p+6),(p+12),(p+14),(p+16) [impossible since one of these would be a multiple of 3]; (p,p+2,p+6),(p+12),(p+14),(p+18) [impossible since one of these would be a multiple of 5]. - _R. J. Mathar_, Jun 15 2013
%D A172456 R. K. Guy, Unsolved Problems in Number Theory, E30.
%H A172456 Amiram Eldar, <a href="/A172456/b172456.txt">Table of n, a(n) for n = 1..1000</a>
%H A172456 G. E. Andrews, <a href="http://www.jstor.org/stable/2318498">MacMahon's prime numbers of measurement</a>, Amer. Math. Monthly, 82 (1975), 922-923.
%H A172456 T. Forbes, <a href="https://mathworld.wolfram.com/PrimeTriplet.html">Prime k-tuplets</a>
%H A172456 R. L. Graham and C. B. A. Peck, <a href="http://www.jstor.org/stable/2315138">Problem E1910</a>, Amer. Math. Monthly, 75 (1968), 80-81.
%H A172456 P. A. MacMahon, <a href="https://www.biodiversitylibrary.org/item/88477#page/679/mode/1up">The prime numbers of measurement on a scale</a>, Proc. Camb. Phil. Soc. 21 (1923), 651-654; reprinted in Coll. Papers I, pp. 797-800.
%H A172456 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeTriplet.html">Prime Triplet</a>.
%e A172456 The first two terms correspond to the sextuples (17,19,23,29,31,37) and (1277,1279,1283,1289,1291,1297).
%p A172456 for n from 1 by 2 to 400000 do; if isprime(n) and isprime(n+2) and isprime(n+6) and isprime(n+12) and isprime(n + 14) and isprime(n+20) then print(n) else fi;od;
%t A172456 Select[Prime[Range[171000]],And@@PrimeQ[{#+2,#+6,#+12,#+14,#+20}]&] (* _Harvey P. Dale_, Jul 23 2011 *)
%t A172456 Select[Prime[Range[171000]],AllTrue[#+{2,6,12,14,20},PrimeQ]&] (* or *) Select[ Partition[Prime[Range[171000]],6,1],Differences[#]=={2,4,6,2,6}&][[All,1]] (* _Harvey P. Dale_, Sep 04 2022 *)
%Y A172456 Initial members of prime quadruples (p, p+2, p+6, p+12): A172454.
%Y A172456 Cf. A073648, A098412.
%K A172456 nonn
%O A172456 1,1
%A A172456 _Michel Lagneau_, Feb 03 2010