A261731 - OEIS

%I #25 Sep 08 2022 08:46:13

%S 1308497,3042491,3042701,7445309,20031101,31572521,44687987,54266291,

%T 141208619,182316521,237416369,357080021,448436321,611641187,

%U 699458411,761126027,774997367,794065967,836452961,915215591,944958941,1009194617,1581935939,1763255561,1871007371

%N Initial member of five twin prime pairs with gap 210 between them.

%C More precisely, primes p such that p+2, p+210, p+212, p+420, p+422, p+630, p+632, p+840, p+842 are all primes.

%C All the terms in this sequence are congruent to 2 (mod 3).

%H K. D. Bajpai and Dana Jacobsen, <a href="/A261731/b261731.txt">Table of n, a(n) for n = 1..10000</a> [first 46 terms from K. D. Bajpai]

%e 1308497 appears in this sequence because: (a) {1308497, 1308499}, {1308707, 1308709}, {1308917, 1308919}, {1309127, 1309129}, and {1309337, 1309339} are five twin prime pairs; (b) the gap between each twin prime pair {1308707 - 1308497} = {1308917-1308707} = {1309127 - 1308917} = {1309337 - 1309127} = 210.

%p select(p -> andmap(isprime, [p, p+2, p+210, p+212, p+420, p+422, p+630, p+632, p+840, p+842]),[seq(p, p=1..2*10^7)]);

%t k = 210; Select[Prime@Range[6*10^7], PrimeQ[# + 2] && PrimeQ[# + k] && PrimeQ[# + k + 2] && PrimeQ[# + 2 k] && PrimeQ[# + 2 k + 2] && PrimeQ[# + 3 k] &&   PrimeQ[# + 3 k + 2] && PrimeQ[# + 4 k] && PrimeQ[# + 4 k + 2] &]

%t Select[Prime[Range[93*10^6]],AllTrue[#+{2,210,212,420,422,630,632,840,842},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Apr 05 2018 *)

%o (PARI) forprime(p= 1,3*10^9, if(isprime(p+2) && isprime(p+210) && isprime(p+212) && isprime(p+420) && isprime(p+422) && isprime(p+630) && isprime(p+632) && isprime(p+840) && isprime(p+842), print1(p,", ")));

%o (Magma) [p: p in PrimesUpTo (100000) | IsPrime(p+2) and IsPrime(p+210) and IsPrime(p+212) and IsPrime(p+420) and IsPrime(p+422) and IsPrime(p+630) and IsPrime(p+632) and IsPrime(p+840) and IsPrime(p+842) ];

%o (Perl) use ntheory ":all"; say for sieve_prime_cluster(1,1e10, 2, 210, 212, 420, 422, 630, 632, 840, 842); # _Dana Jacobsen_, Oct 02 2015

%Y Cf. A077800, A113274, A253624, A261701.

%K nonn

%O 1,1

%A _K. D. Bajpai_, Aug 30 2015