A261362 -id:A261362 - cp's OEIS frontend

A261361 Primes p such that 2*prime(p) + 1 = prime(q) for some prime q.

3, 13, 173, 463, 523, 823, 971, 991, 1291, 1543, 2113, 4003, 4019, 4649, 5923, 6037, 6101, 7649, 10103, 12539, 12841, 17203, 17569, 18013, 21193, 22093, 23321, 25111, 27197, 31231, 32009, 32117, 33349, 34687, 35423, 35449, 37747, 39619, 41729, 41759, 42017, 43237, 43331, 44741, 45841, 50021, 51437, 52489, 55921, 56891

Offset: 1

Zhi-Wei Sun, Aug 16 2015

nonn

Conjecture: The sequence contains infinitely many terms. In general, if a,b,c are positive integers with gcd(a,b) = gcd(a,c) = gcd(b,c) = 1, and a+b+c is even and a is not equal to b, then there are infinitely many prime pairs {p,q} such that a*prime(p) - b*prime(q) = c.
See also A261362 for a stronger conjecture.
Recall that a prime p is called a Sophie Germain prime if 2*p+1 is also prime. A well-known conjecture states that there are infinitely many Sophie Germain primes.

			a(1) = 3 since 3 is a prime, and 2*prime(3)+1 = 2*5+1 = 11 = prime(5) with 5 prime.
a(3) = 173 since 173 is a prime, and 2*prime(173)+1 = 2*1031+1 = 2063 = prime(311) with 311 prime.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A005384, A260886, A260888, A261352, A261354, A261362.

f[n_]:=2*Prime[Prime[n]]+1
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
n=0;Do[If[PQ[f[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,5800}]

A260753 Least positive integer k such that both k and k*n belong to the set {m>0: prime(prime(m))-prime(m)+1 = prime(p) for some prime p}.

Original entry on oeis.org

2, 2, 2279, 5806, 4, 1135, 816, 6556, 725, 2, 1333, 10839, 27, 829, 2279, 2838, 3881, 6540, 2564, 2, 7830, 6540, 27, 4905, 6121, 8220, 316, 1061, 2, 14691, 2, 1168, 738, 4707, 785, 12467, 5492, 1447, 542, 538, 12840, 829, 4732, 5637, 785, 1246, 1198, 433, 58, 573, 26280, 17387, 316, 430, 1198, 4315, 4315, 1479, 4315, 1497

Offset: 1

Zhi-Wei Sun, Aug 18 2015

nonn

Conjecture: For any s and t in the set {1,-1}, every positive rational number r can be written as m/n with m and n in the set {k>0: prime(prime(k))+s*prime(k)+t = prime(p) for some prime p}.

In the case s = -1 and t = 1, the conjecture implies that A261136 has infinitely many terms.

			a(3) = 2279 since  prime(prime(2279))-prime(2279)+1 = prime(20147)-20147+1 = 226553-20146 = 206407 = prime(18503) with 18503 prime, and  prime(prime(2279*3))-prime(2279*3)+1 = prime(68777)-68777+1 = 865757-68776 = 796981 = prime(63737) with 63737 prime.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Zhi-Wei Sun, Checking the conjecture for s = -1, t = 1 and r = a/b (a,b = 1..125)
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A234694, A234695, A236832, A238766, A238878, A261136, A261362.

f[n_]:=Prime[Prime[n]]-Prime[n]+1
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
Do[k=0;Label[bb];k=k+1;If[PQ[f[k]]&&PQ[f[k*n]],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,60}]

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A261361 Primes p such that 2*prime(p) + 1 = prime(q) for some prime q.

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