OFFSET
1,1
COMMENTS
Conjecture: For any positive integer n, each rational number r > 0 can be written as pi(p*n)/pi(q*n) with p and q both prime.
For example, 4/7 = 416/728 = pi(479*6)/pi(919*6) with 479 and 919 both prime.
The conjecture holds trivially for n = 1 since pi(prime(m)*1) = m for all m = 1,2,3,.... Also, the conjecture implies that a(n) exists for any n > 0.
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..300
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
EXAMPLE
a(4) = 67 since pi(67*4) = 56 = 4*14 = 4*pi(11*4) with 11 and 67 both prime.
MATHEMATICA
f[n_]:=PrimePi[n]; Do[k=0; Label[bb]; k=k+1; If[Mod[f[Prime[k]*n], n]>0, Goto[bb]]; Do[If[f[Prime[k]n]==n*f[Prime[j]*n], Goto[aa]]; If[f[Prime[k]n]<n*f[Prime[j]*n], Goto[bb]]; Continue, {j, 1, k}]; Label[aa]; Print[n, " ", Prime[k]]; Continue, {n, 1, 50}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jul 20 2015
STATUS
approved