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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). 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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.

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

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

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.

EXAMPLE

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.

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]

CROSSREFS

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

Sequence in context: A001480 A308117 A270755 * A110917 A070956 A237127

Adjacent sequences:  A305027 A305028 A305029 * A305031 A305032 A305033

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, May 24 2018

STATUS

approved

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Last modified June 19 06:47 EDT 2019. Contains 324218 sequences. (Running on oeis4.)