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

1,1

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

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.

Example

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

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 *)

Crossrefs

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

Cf. A073648, A098412.

Sequence in context: A229833 A362711 A305872 * A022012 A347851 A022546

Adjacent sequences: A172453 A172454 A172455 * A172457 A172458 A172459

Keyword

nonn

Author

Michel Lagneau, Feb 03 2010

Status

approved