OFFSET
1,2
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)
LINKS
Dana Jacobsen, Table of n, a(n) for n = 1..10000
Eric Snyder, An Eratosthenean Sieve for the Prime Decades
EXAMPLE
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.
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 *)
PROG
(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
(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
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Karsten Meyer, Dec 16 2005
EXTENSIONS
Corrected by T. D. Noe, Nov 16 2006
STATUS
approved