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 A239963 Number of triangular numbers below prime(n) which are also primitive roots modulo prime(n). 7
 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, 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, 12, 14, 8, 7, 16, 5, 11, 10, 9, 6, 14, 14, 11, 8, 14, 7 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS 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. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 Z.-W. Sun, New observations on primitive roots modulo primes, arXiv preprint arXiv:1405.0290 [math.NT], 2014. EXAMPLE a(5) = 1 since the triangular number 3*4/2 = 6 is a primitive root modulo prime(5) = 11. a(12) = 1 since the triangular number 5*6/2 = 15 is a primitive root modulo prime(12) = 37. a(19) = 1 since the triangular number 7*8/2 = 28 is a primitive root modulo prime(19) = 67. MATHEMATICA f[k_]:=f[k]=k(k+1)/2 dv[n_]:=dv[n]=Divisors[n] 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}] CROSSREFS Cf. A000040, A000217, A236308, A236966, A237112, A237121, A237594, A239957. Sequence in context: A069239 A321080 A010265 * A084501 A198020 A098037 Adjacent sequences:  A239960 A239961 A239962 * A239964 A239965 A239966 KEYWORD nonn AUTHOR Zhi-Wei Sun, Apr 23 2014 STATUS approved

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Last modified October 19 13:01 EDT 2019. Contains 328222 sequences. (Running on oeis4.)