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A112540
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Numbers m such that (15m-4, 15m-2, 15m+2, 15m+4) is a prime quadruple.
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4
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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
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OFFSET
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1,2
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COMMENTS
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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
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)
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LINKS
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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Select[Range[6610], PrimeQ[15# - 4] && PrimeQ[15# - 2] && PrimeQ[15# + 2] && PrimeQ[15# + 4]&] (* T. D. Noe, Nov 16 2006 *)
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PROG
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(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])
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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