OFFSET
1,1
COMMENTS
Conjecture: 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..105
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
EXAMPLE
a(1) = 5 since prime(5*1)^2-2 = 11^2-2 = 119 = 7*17 = prime(4*1)*prime(7*1).
a(66) = 149073 since prime(149073*66)^2-2 = 176365951^2-2 = 31104948672134399 = 3160879*9840600881 = prime(3448*66)*prime(9840600881*66).
MATHEMATICA
Dv[n_]:=Divisors[Prime[n]^2-2]
L[n_]:=Length[Dv[n]]
P[k_, n_]:=L[k*n]==4&&PrimeQ[Part[Dv[k*n], 2]]&&Mod[PrimePi[Part[Dv[k*n], 2]], n]==0&&PrimeQ[Part[Dv[k*n], 3]]&&Mod[PrimePi[Part[Dv[k*n], 3]], n]==0
Do[k=0; Label[bb]; k=k+1; If[P[k, n], Goto[aa]]; Goto[bb]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 50}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jul 15 2015
STATUS
approved