OFFSET
1,2
COMMENTS
Conjecture: (i) Any positive rational number r can be written as m/n, where m and n are positive integers such that (prime(prime(m))+prime(prime(n)))/2 = prime(p) for some prime p.
(ii) Any positive rational number r <= 1 can be written as m/n, where m and n are positive integers such that prime(prime(m)),prime(prime(n)), prime(p) and prime(q) form a 4-term arithmetic progression for some pair of primes p and q.
(iii) For any integer k > 2, the set {prime(p): p is prime} contains infinitely many nontrivial k-term arithmetic progressions.
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..100
B. Green and T. Tao, The primes contain arbitrary long arithmetic progressions, Annals of Math. 167(2008), 481-547.
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
EXAMPLE
a(2) = 5109 since prime(prime(5109)) = 608591, prime(prime(5109*2)) = 1401791, prime(162343) = 2194991, and prime(216023) = 2988191 form a 4-term arithmetic progression with 162343 and 216023 both prime.
MATHEMATICA
f[n_]:=Prime[Prime[n]]
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
Do[k=0; Label[bb]; k=k+1; If[PQ[2*f[k*n]-f[k]]&&PQ[3*f[k*n]-2*f[k]], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 50}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 20 2015
STATUS
approved