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A259484 Smallest nonprime number having least positive primitive root n, or 0 if no such root exists.

%I A259484 #23 Feb 16 2025 08:33:26
%S A259484 1,0,9,4,0,6,1681,22,0,0,97969,118,16900321,914,1062961,542,0,262,
%T A259484 2827367929,382
%N A259484 Smallest nonprime number having least positive primitive root n, or 0 if no such root exists.
%C A259484 The value 0 at indices 4, 8, 9, 16, ..., says 0 has no primitive roots (A001597), but the 0 at index 1 says 1 has a primitive root of 0, the only real 0 in the sequence.
%C A259484 a(n) cannot be 2, 4, the odd power of a prime or twice the odd power of a prime.
%C A259484 Conjecture: each odd-indexed value will be populated before either of its even-indexed neighbors.
%D A259484 A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots. Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968, p. XLIV.
%H A259484 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimitiveRoot.html"> Primitive Root.</a>
%F A259484 a(n) = 0 if n is a perfect power (A001597).
%e A259484 a(2) = 9 because the least primitive root of the nonprime number 9 is 2 and no nonprime less than 9 meets this criterion.
%t A259484 smallestPrimitiveRoot[n_ /; n <= 1] = 0; smallestPrimitiveRoot[n_] := Block[{pr = PrimitiveRoot[n], g}, If[ !NumericQ[pr], g = 0, g = 1; While[g <= pr, If[ CoprimeQ[g, n] && MultiplicativeOrder[g, n] == EulerPhi[n], Break[]]; g++]]; g]; (* This part of the code is from _Jean-François Alcover_ as found in A046145, Feb 15 2012 *)
%t A259484 t = Table[-1, {1000}];  ppQ[n_] := GCD @@ Last /@ FactorInteger@ n > 1; ppQ[1] = True; k = 1; While[ k < 1001, If[ ppQ@ k, t[[k]] = 0]; k++]; k = 1; While[k < 200000001, If[ !PrimeQ[k], a = smallestPrimitiveRoot[k]; If[ t[[a]] == -1, t[[a]] = k]]; k++]; t
%Y A259484 Cf. A001597, A023048, A046145, A214158.
%K A259484 nonn,hard,more
%O A259484 0,3
%A A259484 _Robert G. Wilson v_, Jun 28 2015
%E A259484 a(18)-a(19) from _Robert G. Wilson v_, Sep 26 2015