A260947 Least positive integer k < prime(n) such that k = i + j for some 0 < i < j for which i,j,k and i*j*k are all primitive roots modulo prime(n), or 0 if no such k exists.

0, 0, 0, 0, 8, 0, 10, 13, 15, 10, 24, 15, 13, 29, 15, 8, 8, 17, 13, 28, 20, 35, 8, 19, 15, 15, 11, 7, 24, 17, 29, 8, 24, 15, 10, 13, 20, 18, 15, 5, 8, 28, 47, 15, 5, 41, 29, 11, 8, 31, 17, 21, 51, 24, 10, 15, 10, 21, 11, 15

Offset

1,5

Comments

Conjecture: a(n) > 0 for all n > 6. In other words, for any prime p > 13, there are three distinct elements a,b,c of {1,...,p-1} with a+b = c such that a,b,c and a*b*c are all primitive roots modulo p.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(5) = 8 since 2 + 6 = 8, and the four numbers 2, 6, 8 and 2*6*8=96 are all primitive roots modulo prime(5) = 11.

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
Dv[n_]:=Divisors[Prime[n]-1]
Do[Do[Do[If[Mod[k^(Part[Dv[n], t]), Prime[n]]==1, Goto[bb]], {t, 1, Length[Dv[n]]-1}]; Do[Do[If[Mod[j^(Part[Dv[n], t]), Prime[n]]==1||Mod[(k-j)^(Part[Dv[n], t]), Prime[n]]==1||Mod[((k-j)j*k)^(Part[Dv[n], t]), Prime[n]]==1, Goto[cc]]; Continue, {t, 1, Length[Dv[n]]-1}]; Print[n, " ", k]; Goto[aa]; Label[cc]; Continue, {j, 1, (k-1)/2}]; Label[bb]; Continue, {k, 1, Prime[n]-1}]; Print[n, " ", 0]; Label[aa]; Continue, {n, 1, 60}]

Crossrefs

Cf. A000040, A260946.

Sequence in context: A176460 A320378 A228412 * A070489 A112268 A077062

Adjacent sequences: A260944 A260945 A260946 * A260948 A260949 A260950

Keyword

nonn

Author

Zhi-Wei Sun, Aug 05 2015

Status

approved