OFFSET
1,1
COMMENTS
Conjecture: Any positive rational number r can be written as m/n with m and n in the set {k>0: prime(k)+2 is prime with prime(prime(k)+2) = prime(prime(k))+6}.
This implies that there are infinitely many twin prime pairs {p, p+2} with prime(p+2) - prime(p) = 6.
Note that if prime(n+2)-prime(n) = 6 then prime(n+1)-prime(n) = 2 or 4.
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
Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..20
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
EXAMPLE
a(1) = 2 since 2*1 = 2, and prime(2)+2 = 3+2 = 5 is prime with prime(5)-prime(3) = 11-5 = 6.
a(2) = 891 since prime(891)+2 = 6947 + 2 = 6949 is prime with prime(6949)-prime(6947) = 70123-70117 = 6, and prime(891*2)+2 = 15269 + 2 = 15271 is prime with prime(15271)-prime(15269) = 167119-167113 = 6.
MATHEMATICA
f[n_]:=Prime[n]
PQ[k_]:=PrimeQ[f[k]+2]&&f[f[k]+2]-f[f[k]]==6
Do[k=0; Label[bb]; k=k+1; If[PQ[k]&&PQ[k*n], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 40}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 23 2015
STATUS
approved