A236482 -id:A236482 - 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}]

A236484 Primes p with p + 2, prime(p) + 2, prime(prime(p)) + 2, prime(prime(prime(p))) + 2, prime(prime(prime(prime(p)))) + 2 all prime.

Original entry on oeis.org

2371709, 3406727, 8890667, 45809639, 57219497, 58674437, 73793831, 78934589, 159935561, 207223409

Offset: 1

Zhi-Wei Sun, Jan 27 2014

nonn

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

			a(1) =  2371709 with  2371709, 2371709 + 2 = 2371711, prime(2371709) + 2 = 38917889 + 2 = 38917891, prime(38917889) + 2 =  754394519 + 2 = 754394521, prime(754394519) + 2 = 16978533527 + 2 = 16978533529 and prime(16978533527) + 2 =  437397516929 + 2 = 437397516931 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10

Cf. A000040, A001359, A006512, A236481, A236482.

p[n_]:=p[n]=PrimeQ[n+2]&&PrimeQ[Prime[n]+2]&&PrimeQ[Prime[Prime[n]]+2]&&PrimeQ[Prime[Prime[Prime[n]]]+2]&&PrimeQ[Prime[Prime[Prime[Prime[n]]]]+2]
n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10^7}]

A237283 Primes p with prime(prime(p)) + 2 also prime.

Original entry on oeis.org

2, 3, 7, 13, 23, 29, 59, 71, 103, 193, 257, 271, 281, 311, 317, 389, 433, 439, 463, 569, 577, 619, 673, 683, 691, 797, 811, 857, 859, 887, 1031, 1069, 1109, 1129, 1153, 1229, 1307, 1597, 1613, 1867, 1949, 1951, 2069, 2297, 2477, 2551, 2621, 2657, 2699, 2753

Offset: 1

Zhi-Wei Sun, Feb 05 2014

nonn

This sequence is interesting because of the conjecture in A237253.

A236481, A236482 and A236484 are subsequences of the sequence.

			a(1) = 2 since 2 and prime(prime(2)) + 2 = prime(3) + 2 = 7 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A001359, A006512, A096478, A218829, A236457, A236458, A236467, A236481, A236482, A236484, A236568, A237253.

n=0;Do[If[PrimeQ[Prime[Prime[Prime[k]]]+2],n=n+1;Print[n," ",Prime[k]]],{k,1,1000}]
Select[Prime[Range[500]],PrimeQ[Prime[Prime[#]]+2]&] (* Harvey P. Dale, May 30 2018 *)

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

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A236484 Primes p with p + 2, prime(p) + 2, prime(prime(p)) + 2, prime(prime(prime(p))) + 2, prime(prime(prime(prime(p)))) + 2 all prime.

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A237283 Primes p with prime(prime(p)) + 2 also prime.

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