A261701 - OEIS

%I #28 Sep 08 2022 08:46:13

%S 599,3917,5021,37361,48779,81929,93281,97157,263399,433049,783149,

%T 821801,906119,908669,1197197,1308497,1308707,1379237,1464809,1908449,

%U 2036861,2341979,2408561,2760671,2804309,3042491,3042701,3042911,3198197,4090649,4543991,5543927

%N Initial member of four 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 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="/A261701/b261701.txt">Table of n, a(n) for n = 1..10000</a> [first 865 terms from K. D. Bajpai]

%H Luis Rodriguez and Carlos Rivera, <a href="http://www.primepuzzles.net/conjectures/conj_066.htm">Gaps between consecutive twin pairs</a>, The Prime Puzzles and Problems Connection.

%e 599 appears in the sequence because: (a) {599,601}, {809, 811}, {1019, 1021}, {1229, 1231} are four (not consecutive) twin prime pairs; (b) the gap between each twin prime pair (809 - 599) = (1019 -  809) = (1229 - 1019) = 210.

%p select(p -> andmap(isprime, [p, p+2, p+210, p+212, p+420, p+422, p+630, p+632]),[seq(p, p=1..10^5)]);

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

%t Select[Prime[Range[400000]],AllTrue[#+{2,210,212,420,422,630,632},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Jul 17 2019 *)

%o (PARI) forprime(p= 1, 100000, if(isprime(p+2) && isprime(p+210) && isprime(p+212) && isprime(p+420) && isprime(p+422) && isprime(p+630) && isprime(p+632), 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) ];

%o (Perl) use ntheory ":all"; say join ", ", grep { is_prime($_+210) && is_prime($_+212) && is_prime($_+420) && is_prime($_+422) && is_prime($_+630) && is_prime($_+632) } @{twin_primes(1e8)}; # _Dana Jacobsen_, Sep 02 2015

%o (Perl) use ntheory ":all"; say for sieve_prime_cluster(1, 1e8, 2, 210, 212, 420, 422, 630, 632); # _Dana Jacobsen_, Oct 03 2015

%Y Cf. A001359 (twin primes), A077800, A113274, A253624.

%K nonn

%O 1,1

%A _K. D. Bajpai_, Aug 28 2015