A305030 - OEIS

%I #16 May 25 2018 06:02:07

%S 0,1,2,1,3,2,3,2,2,5,4,2,4,3,3,7,3,3,3,3,2,2,6,4,7,9,4,8,2,5,3,8,7,9,

%T 9,4,3,6,5,9,10,5,5,5,8,3,5,7,5,4,6,4,2,5,8,7,14,6,4,9,8,7,3,5,6,11,6,

%U 5,13,8,8,8,8,4,8,7,14,6,7,7,8,8,8,5,3,8,6,5,9,5

%N Number of pairs {k, m} with 0 <= k <= m such that binomial(2k,k) + binomial(2m,m) is not only a primitive root modulo prime(n) but also smaller than prime(n).

%C Conjecture 1: a(n) > 0 for all n > 1. In other words, any odd prime p has a primitive root g < p which is the sum of two central binomial coefficients.

%C Conjecture 2: Each odd prime p has a primitive root g < p which is the sum of two Catalan numbers.

%C We have verified Conjecture 1 for all odd primes p < 10^9.

%H Zhi-Wei Sun, <a href="/A305030/b305030.txt">Table of n, a(n) for n = 1..50000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1405.0290">New observations on primitive roots modulo primes</a>, arXiv:1405.0290 [math.NT], 2014.

%H Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/160p.pdf">Problems on combinatorial properties of primes</a>, 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.

%e a(2) = 1 with binomial(2*0,0) + binomial(2*0,0) = 2 a primitive root modulo prime(2) = 3.

%e a(3) = 2 with binomial(2*0,0) + binomial(2*0,0) = 2 and binomial(2*0,0) + binomial(2*1,1) = 3 primitive roots modulo prime(3) = 5.

%e a(4) = 1 with binomial(2*0,0) + binomial(2*1,1) = 3 a primitive root modulo prime(4) = 7.

%e a(29) = 2 with binomial(2*3,3) + binomial(2*3,3) = 40 and binomial(2*1,1) + binomial(2*4,4) = 72 primitive roots modulo prime(29) = 109.

%t p[n_]:=p[n]=Prime[n];

%t Dv[n_]:=Dv[n]=Divisors[n];

%t gp[g_,p_]:=gp[g,p]=Mod[g,p]>0&&Sum[Boole[PowerMod[g,Dv[p-1][[k]],p]==1],{k,1,Length[Dv[p-1]]-1}]==0;

%t tab={};Do[r=0;a=0;Label[aa];If[Binomial[2a,a]>=p[n],Goto[cc]];b=0;Label[bb];If[b>a||Binomial[2b,b]>=p[n]-Binomial[2a,a],a=a+1;Goto[aa]];

%t If[gp[Binomial[2a,a]+Binomial[2b,b],p[n]],r=r+1];b=b+1;Goto[bb];Label[cc];tab=Append[tab,r],{n,1,90}];Print[tab]

%Y A000040, A000108, A000984, A303540, A239957, A241476, A241504, A241516, A305048.

%K nonn

%O 1,3

%A _Zhi-Wei Sun_, May 24 2018