A241516 Least positive primitive root g < prime(n) modulo prime(n) which is also a partition number given by A000041, or 0 if such a number g does not exist.

1, 2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, 7, 3, 5, 2, 2, 2, 2, 7, 5, 3, 2, 3, 5, 2, 5, 2, 11, 3, 3, 2, 3, 2, 2, 7, 5, 2, 5, 2, 2, 2, 22, 5, 2, 3, 2, 3, 2, 7, 3, 7, 7, 11, 3, 5, 2, 15, 5, 3, 3, 2, 5, 22, 15, 2, 3, 15, 2, 2, 3, 7, 11, 2, 2, 5, 2, 5, 3, 22

Offset: 1

Author

Zhi-Wei Sun, Apr 24 2014

Keywords

• nonn

Comments

According to the conjecture in A241504, a(n) should be always positive.

Examples

			a(4) = 3 since 3 = A000041(3) is a primitive root modulo prime(4) = 7, but neither 1 = A000041(1) nor 2 = A000041(2) is.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Crossrefs

Cf. A000040, A000041, A237121, A239957, A239963, A241476, A241492, A241504.

Programs

Mathematica

f[k_]:=PartitionsP[k]
dv[n_]:=Divisors[n]
Do[Do[If[f[k]>Prime[n]-1,Goto[cc]];Do[If[Mod[f[k]^(Part[dv[Prime[n]-1],i]),Prime[n]]==1,Goto[aa]],{i,1,Length[dv[Prime[n]-1]]-1}];Print[n," ",PartitionsP[k]];Goto[bb];Label[aa];Continue,{k,1,Prime[n]-1}];Label[cc];Print[Prime[n]," ",0];Label[bb];Continue,{n,1,80}]