OFFSET
1,4
COMMENTS
Conjecture: a(n) > 0 for all n > 1.
This implies that there are infinitely many positive integers k which is a primitive root modulo prime(k).
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..8000
EXAMPLE
a(6) = 1 since 6 = 3 + 3 with 3 a primitive root modulo prime(3) = 5.
a(7) = 1 since 7 = 1 + 6 with 1 a primitive root modulo prime(1) = 2 and 6 a primitive root modulo prime(6) = 13.
a(15) = 1 since 15 = 2 + 13 with 2 a primitive root modulo prime(2) = 3 and 13 a primitive root modulo prime(13) = 41.
a(38) = 1 since 38 = 10 + 28 with 10 a primitive root modulo prime(10) = 29 and 28 a primitive root modulo prime(28) = 107.
a(53) = 1 since 53 = 3 + 50 with 3 a primitive root modulo prime(3) = 5 and 50 a primitive root modulo prime(50) = 229.
MATHEMATICA
dv[n_]:=Divisors[n]
Do[m=0; Do[Do[If[Mod[k^(Part[dv[Prime[k]-1], i]), Prime[k]]==1, Goto[aa]], {i, 1, Length[dv[Prime[k]-1]]-1}]; Do[If[Mod[(n-k)^(Part[dv[Prime[n-k]-1], j]), Prime[n-k]]==1, Goto[aa]], {j, 1, Length[dv[Prime[n-k]-1]]-1}]; m=m+1; Label[aa]; Continue, {k, 1, n/2}]; Print[n, " ", m]; Continue, {n, 1, 80}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 21 2014
STATUS
approved