A112540 Numbers m such that (15m-4, 15m-2, 15m+2, 15m+4) is a prime quadruple.

1, 7, 13, 55, 99, 125, 139, 217, 231, 377, 629, 867, 1043, 1049, 1071, 1203, 1261, 1295, 1401, 1485, 1687, 2115, 2323, 2919, 3423, 3689, 4199, 4481, 4633, 4815, 5151, 5313, 5403, 5515, 5921, 6523, 6609, 6741, 7323, 7769, 7953, 8147, 9031, 9611, 10485, 11047

Offset: 1

Author

Karsten Meyer, Dec 16 2005

Keywords

• nonn
• easy

Comments

Also (4p + 16)/60 such that (p, p+2, p+6 and p+8) is a prime quadruple for p >= 11. - Michel Lagneau, Jul 02 2012

The density of these four-prime groups is approximately equal to (log x)^-3.45 (but not (log x)^-4). - Xueshi Gao, Jun 01 2014

All of the terms of this sequence are either 1, 7 or 13 modulo 14. - Rodolfo Ruiz-Huidobro, Dec 27 2019

From Eric Snyder, Jun 23 2021: (Start)

Building on the comment of R. Ruiz-Huidobro above, all terms of the sequence are congruent to one of {-1, 0 ,1} (mod 7). The appearance of mod 14 stems from the fact that all entries in this list must be odd. Equivalently, a(n) cannot be +- 2 or +- 3 (mod 7). This can be generalized for all larger primes:

All primes p >= 7 can be expressed as 15k +- q in a least absolute residue system, with q in {2, 4} if k is odd, and q in {1,7} if k is even.

For all primes 15k +- q >= 7, four residues +-r, +-s (mod p) exist such that, if for any p >= 7, [(m == +- r (mod p) or m == +- s (mod p)), and (m != k)], then m is not in this sequence. For the different values of p = 15k +- q, the values of +-r and +-s are as follows:

For p = 15k +- 1 (k even), r = +- 2k, s = +- 4k

For p = 15k +- 2 (k odd), r = +- k, s = +- 2k

For p = 15k + 4 (k odd), r = +- k, s = +- (7k + 2)

For p = 15k - 4 (k odd), r = +- k, s = +- (7k - 2)

For p = 15k + 7 (k even), r = +- (4k + 2), s = +- (8k + 4)

For p = 15k - 7 (k even), r = +- (4k - 2), s = +- (8k - 4)

These can be used to create an Eratosthenes-like sieve for the prime decades. (End)

Examples

			m = 7 yields the quadruple (15*7 - 4 = 101, 15*7 - 2 = 103, 15*7 + 2 = 107, 15*7 + 4 = 109), so 7 is a term of the sequence.
		

Links

• Dana Jacobsen, Table of n, a(n) for n = 1..10000
• Eric Snyder, An Eratosthenean Sieve for the Prime Decades

Crossrefs

Cf. A007530, A014561.

Programs

Magma

[n: n in [0..2*10^4] | IsPrime(15*n-4) and IsPrime(15*n-2) and IsPrime(15*n+2) and IsPrime(15*n+4)]; // Vincenzo Librandi, Dec 28 2015

Maple

A112540:=n->`if`(isprime(15*n-4) and isprime(15*n-2) and isprime(15*n+2) and isprime(15*n+4),n,NULL); seq(A112540(n), n=1..20000); # Wesley Ivan Hurt, Jul 26 2014

Mathematica

Select[Range[6610], PrimeQ[15# - 4] && PrimeQ[15# - 2] && PrimeQ[15# + 2] && PrimeQ[15# + 4]&] (* T. D. Noe, Nov 16 2006 *)

PARI

for(n=1, 1e4, if(isprime(15*n-4) && isprime(15*n-2) && isprime(15*n+2) && isprime(15*n+4), print1(n, ", "))) \\ Felix Fröhlich, Jul 26 2014

Perl

use ntheory ":all"; say for map { (4*$+16)/60 } sieve_prime_cluster(11,15*10000, 2,6,8); # _Dana Jacobsen, Dec 15 2015

Python

from sympy import isprime
def ok(m): return all(isprime(15*m+k) for k in [-4, -2, 2, 4])
print(list(filter(ok, range(11111)))) # Michael S. Branicky, Jun 24 2021

Extensions

Corrected by T. D. Noe, Nov 16 2006