

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
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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 wellknown conjecture states that there are infinitely many Sophie Germain primes.


REFERENCES

ZhiWei 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 ChinaJapan Seminar (Fukuoka, Oct. 28  Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169187.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei 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

ZhiWei Sun, Aug 16 2015


STATUS

approved



