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A261361 Primes p such that 2*prime(p) + 1 = prime(q) for some prime q. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

REFERENCES

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.

LINKS

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.

EXAMPLE

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.

MATHEMATICA

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}]

CROSSREFS

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

Sequence in context: A290769 A213794 A239979 * A114317 A168320 A323134

Adjacent sequences:  A261358 A261359 A261360 * A261362 A261363 A261364

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Aug 16 2015

STATUS

approved

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Last modified October 18 13:31 EDT 2019. Contains 328161 sequences. (Running on oeis4.)