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(prime(m))) + prime(prime(prime(n))))/2 = prime(prime(p)) for some prime p.
(ii) Let p(1,n) = prime(n), and p(m+1,n) = p(m,prime(n)) for m,n = 1,2,3,.... Then, for any integers m > 0 and k > 2, the sequence p(m,n) (n = 1,2,3,...) contains infinitely many nontrivial k-term arithmetic progressions.
(iii) Let m be any positive integer. Then the sequence p(m,n)^(1/n) (n = 1,2,3,...) is strictly decreasing. Also, for any relatively prime integers q > 0 and r, there are infinitely many n > 0 such that p(m,n) == r (mod q).
Note that part (ii) of the conjecture extends the Green-Tao theorem and the third part of the conjecture in A261462. Also, part (iii) in the case m = 1 reduces to the Firoozbakht conjecture on primes and Dirichlet's theorem on primes in 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..200
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) = 755 since prime(prime(prime(755))) + prime(prime(prime(755*2))) = prime(prime(5741)) + prime(prime(12641)) = prime(56611) + prime(135671) = 700897 + 1808581 = 2*1254739 = 2*prime(96797) = 2*prime(prime(9319)) with 9319 prime.
MATHEMATICA
f[n_]:=Prime[Prime[Prime[n]]]
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]&&PrimeQ[PrimePi[PrimePi[p]]]
Do[k=0; Label[bb]; k=k+1; If[PQ[(f[k]+f[k*n])/2], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 60}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 25 2015
STATUS
approved