A261701 Initial member of four twin prime pairs with gap 210 between them.

599, 3917, 5021, 37361, 48779, 81929, 93281, 97157, 263399, 433049, 783149, 821801, 906119, 908669, 1197197, 1308497, 1308707, 1379237, 1464809, 1908449, 2036861, 2341979, 2408561, 2760671, 2804309, 3042491, 3042701, 3042911, 3198197, 4090649, 4543991, 5543927

Offset

1,1

Comments

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

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

Links

K. D. Bajpai and Dana Jacobsen, Table of n, a(n) for n = 1..10000 [first 865 terms from K. D. Bajpai]

Luis Rodriguez and Carlos Rivera, Gaps between consecutive twin pairs, The Prime Puzzles and Problems Connection.

Example

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.

Maple

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)]);

Mathematica

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] &]
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 *)

Prog

(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, ", ")));
(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) ];
(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
(Perl) use ntheory ":all"; say for sieve_prime_cluster(1, 1e8, 2, 210, 212, 420, 422, 630, 632); # Dana Jacobsen, Oct 03 2015

Crossrefs

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

Sequence in context: A106762 A158277 A250937 * A362324 A172244 A090222

Adjacent sequences: A261698 A261699 A261700 * A261702 A261703 A261704

Keyword

nonn

Author

K. D. Bajpai, Aug 28 2015

Status

approved