A236480 -id:A236480 - cp's OEIS frontend

A236481 Primes p with p + 2, prime(p) + 2 and prime(prime(p)) + 2 all prime.

3, 1949, 4217, 8219, 9929, 22091, 23537, 28097, 38711, 41609, 50051, 60899, 68111, 72227, 74159, 79631, 115151, 122399, 127679, 150959, 155537, 266687, 267611, 270551, 271499, 284741, 306347, 428297, 433661, 444287

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

Conjecture: For any positive integer m, there are infinitely many chains p(1) < p(2) < ... < p(m) of m primes with p(k) + 2 prime for all k = 1,...,m such that p(k + 1) = prime(p(k)) for every 0 < k < m.

			a(1) = 3 since 3, 3 + 2 = 5, prime(3) + 2 = 7 and prime(prime(3)) + 2 = prime(5) + 2 = 13 are all prime, but 2 + 2 = 4 is composite.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Zhi-Wei Sun, A new kind of prime chains, a message to Number Theory List, Jan. 20, 2014.

Cf. A000010, A000040, A001359, A006512, A235925, A236066, A236457, A236458, A236468, A236470, A236480, A236482, A236484.

p[n_]:=p[n]=PrimeQ[n+2]&&PrimeQ[Prime[n]+2]&&PrimeQ[Prime[Prime[n]]+2]
n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10^6}]

A236482 Primes p with p + 2, prime(p) + 2, prime(prime(p)) + 2 and prime(prime(prime(p))) + 2 all prime.

Original entry on oeis.org

41609, 1119047, 1928621, 2348579, 2371709, 3406727, 4098569, 4204817, 4438997, 5561819, 6161159, 6293297, 8236439, 8736701, 8890667, 8951387, 9231329, 9390077, 10492457, 10619897, 11255729, 11514719, 11769479, 11920661, 12316697

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

According to the general conjecture in A236481, this sequence should have infinitely many terms.

			a(1) = 41609 with 41609, 41609 + 2 = 41611, prime(41609) + 2 = 500909 + 2 = 500911, prime(500909) + 2 = 7382957 + 2 = 7382959 and prime(7382957) + 2 = 130090109 + 2 = 130090111 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100

Cf. A000040, A001359, A006512, A236457, A236458, A236467, A236480, A236481, A236484.

p[n_]:=p[n]=PrimeQ[n+2]&&PrimeQ[Prime[n]+2]&&PrimeQ[Prime[Prime[n]]+2]&&PrimeQ[Prime[Prime[Prime[n]]]+2]
n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10^6}]

cp's OEIS Frontend

A236481 Primes p with p + 2, prime(p) + 2 and prime(prime(p)) + 2 all prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica