OFFSET
1,2
COMMENTS
Conjecture: a(n) > 0 for all n > 0.
This is stronger than the well-known conjecture that there are infinitely many primes of the form x^2+1 with x an integer.
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..2000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
EXAMPLE
a(1) = 1 since prime(1*1)-1 = 2-1 = 1^2.
a(2) = 6 since prime(6*2)-1 = 37-1 = 6^2.
MATHEMATICA
SQ[n_]:=IntegerQ[Sqrt[n]]
Do[k=0; Label[bb]; k=k+1; If[SQ[Prime[k*n]-1], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 70}]
lpi[n_]:=Module[{k=1}, While[!IntegerQ[Sqrt[Prime[k*n]-1]], k++]; k]; Array[ lpi, 70] (* Harvey P. Dale, Apr 18 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jul 04 2015
STATUS
approved