A305030 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).
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, 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, 5, 13, 8, 8, 8, 8, 4, 8, 7, 14, 6, 7, 7, 8, 8, 8, 5, 3, 8, 6, 5, 9, 5
Offset: 1
Keywords
Examples
a(2) = 1 with binomial(2*0,0) + binomial(2*0,0) = 2 a primitive root modulo prime(2) = 3. 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. a(4) = 1 with binomial(2*0,0) + binomial(2*1,1) = 3 a primitive root modulo prime(4) = 7. 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.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..50000
- Zhi-Wei Sun, New observations on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.
- 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.
Programs
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Mathematica
p[n_]:=p[n]=Prime[n]; Dv[n_]:=Dv[n]=Divisors[n]; 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; 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]]; 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]
Comments