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A239963 Number of triangular numbers below prime(n) which are also primitive roots modulo prime(n).

%I A239963 #24 Aug 05 2019 02:49:07
%S A239963 1,0,1,1,1,1,3,3,3,4,2,1,3,2,3,3,3,3,1,3,3,4,5,5,3,5,4,9,3,7,6,4,7,3,
%T A239963 9,3,7,5,10,9,10,9,5,10,7,7,2,5,8,6,8,7,6,6,12,10,8,9,7,10,8,11,6,6,
%U A239963 12,14,8,7,16,5,11,10,9,6,14,14,11,8,14,7
%N A239963 Number of triangular numbers below prime(n) which are also primitive roots modulo prime(n).
%C A239963 Conjecture: a(n) > 0 for all n > 2. In other words, for any prime p > 3, there is a primitive root 0 < g < p of the form k*(k+1)/2, where k is a positive integer.
%H A239963 Zhi-Wei Sun, <a href="/A239963/b239963.txt">Table of n, a(n) for n = 1..10000</a>
%H A239963 Z.-W. Sun, <a href="http://arxiv.org/abs/1405.0290">New observations on primitive roots modulo primes</a>, arXiv preprint arXiv:1405.0290 [math.NT], 2014.
%e A239963 a(5) = 1 since the triangular number 3*4/2 = 6 is a primitive root modulo prime(5) = 11.
%e A239963 a(12) = 1 since the triangular number 5*6/2 = 15 is a primitive root modulo prime(12) = 37.
%e A239963 a(19) = 1 since the triangular number 7*8/2 = 28 is a primitive root modulo prime(19) = 67.
%t A239963 f[k_]:=f[k]=k(k+1)/2
%t A239963 dv[n_]:=dv[n]=Divisors[n]
%t A239963 Do[m=0;Do[Do[If[Mod[f[k]^(Part[dv[Prime[n]-1],i]),Prime[n]]==1,Goto[aa]],{i,1,Length[dv[Prime[n]-1]]-1}];m=m+1;Label[aa];Continue,{k,1,(Sqrt[8Prime[n]-7]-1)/2}];Print[n," ",m];Continue,{n,1,80}]
%Y A239963 Cf. A000040, A000217, A236308, A236966, A237112, A237121, A237594, A239957.
%K A239963 nonn
%O A239963 1,7
%A A239963 _Zhi-Wei Sun_, Apr 23 2014