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

17, 1277, 1607, 3527, 4637, 71327, 97367, 113147, 191447, 290657, 312197, 416387, 418337, 421697, 450797, 566537, 795647, 886967, 922067, 1090877, 1179317, 1300127, 1464257, 1632467, 1749257, 1866857, 1901357, 2073347, 2322107

Offset: 1

Author

Michel Lagneau, Feb 03 2010

Keywords

• nonn

Comments

The last digit of each of these prime numbers is 7.

Subsequence of A078946.

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

Examples

			The first two terms correspond to the sextuples (17,19,23,29,31,37) and (1277,1279,1283,1289,1291,1297).
		

References

• R. K. Guy, Unsolved Problems in Number Theory, E30.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000
• G. E. Andrews, MacMahon's prime numbers of measurement, Amer. Math. Monthly, 82 (1975), 922-923.
• T. Forbes, Prime k-tuplets
• R. L. Graham and C. B. A. Peck, Problem E1910, Amer. Math. Monthly, 75 (1968), 80-81.
• P. A. MacMahon, The prime numbers of measurement on a scale, Proc. Camb. Phil. Soc. 21 (1923), 651-654; reprinted in Coll. Papers I, pp. 797-800.
• Eric Weisstein's World of Mathematics, Prime Triplet.

Crossrefs

Initial members of prime quadruples (p, p+2, p+6, p+12): A172454.

Cf. A073648, A098412.

Programs

Maple

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;

Mathematica

Select[Prime[Range[171000]],And@@PrimeQ[{#+2,#+6,#+12,#+14,#+20}]&] (* Harvey P. Dale, Jul 23 2011 *)
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 *)