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Numbers n such that (6k-1) for k=n, n+1, n+2, n+3 are all primes with no primes of the form (6k+1) in between.
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%I #98 Feb 25 2024 06:00:12

%S 42,897,1052,2107,2242,2457,2632,2912,3887,4362,9347,10367,12587,

%T 13132,13797,14072,14897,15737,15877,17452,19292,20092,20167,25677,

%U 27042,27307,29967,30842,31227,31837,34337,35742,37052,37772,40587,40957,41672,42147,43687,44192

%N Numbers n such that (6k-1) for k=n, n+1, n+2, n+3 are all primes with no primes of the form (6k+1) in between.

%C This sequence of numbers is formed by positive integers k that make 6k-1, 6k+5, 6k+11 and 6k+17 prime numbers with no primes of the form 6k+1 in between. All prime numbers larger than 3 can be expressed as 6k-1 or 6k+1. Not all positive k make a prime number. Only certain k under certain conditions can make 6k-1 or 6k+1 prime.

%C All terms are == 2 (mod 5). - _Robert G. Wilson v_, Dec 14 2017

%H Robert Israel, <a href="/A296011/b296011.txt">Table of n, a(n) for n = 1..10000</a>

%e 42 is in the sequence because 6*42-1=251, 6*43-1=257, 6*44-1=263, 6*45-1=296 are prime and there are no other primes in between, i.e., 6*42+1=253=11*23, 6*43+1=259=7*37, 6*44+1=265=5*53 are not primes.

%t Block[{nn = 50000, s}, s = Select[Prime@ Range@ PrimePi[6 (nn + 3) + 1], Divisible[(# - 1), 6] &]; Select[Range@ nn, And[AllTrue[#, PrimeQ], Count[s, q_ /; First[#] < q < Last@ #] == 0] &@ Map[6 # - 1 &, # + Range[0, 3]] &]] (* _Michael De Vlieger_, Dec 06 2017 *)

%t fQ[n_] := Block[{p = {6n -1, 6n +5, 6n +11, 6n +17}}, Union@ PrimeQ@ p == {True} && NextPrime[6n -1, 3] == 6n +17]; Select[Range@50000, fQ] (* _Robert G. Wilson v_, Dec 14 2017 *)

%o (Sage)

%o a, b, c, d = 2, 3, 5, 7; R = []

%o for p in primes(10**5):

%o if a % 6 + 1 == b - a == c - b == d - c == 6:

%o R.append((a+1)//6)

%o a, b, c, d = b, c, d, p

%o R # _Peter Luschny_, Jan 08 2018

%o (PARI) isok(n) = isprime(6*n-1) && isprime(6*n+5) && isprime(6*n+11) && isprime(6*n+17) && ((primepi(6*n+17) - primepi(6*n-1)) == 3); \\ _Michel Marcus_, Dec 11 2017

%Y Cf. A090839.

%Y Equals A090836+1.

%K nonn,hear

%O 1,1

%A _Pedro Caceres_, Dec 02 2017