OFFSET
0,3
COMMENTS
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.
a(n) cannot be 2, 4, the odd power of a prime or twice the odd power of a prime.
Conjecture: each odd-indexed value will be populated before either of its even-indexed neighbors.
REFERENCES
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.
LINKS
Eric Weisstein's World of Mathematics, Primitive Root.
FORMULA
a(n) = 0 if n is a perfect power (A001597).
EXAMPLE
a(2) = 9 because the least primitive root of the nonprime number 9 is 2 and no nonprime less than 9 meets this criterion.
MATHEMATICA
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 = 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
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Robert G. Wilson v, Jun 28 2015
EXTENSIONS
a(18)-a(19) from Robert G. Wilson v, Sep 26 2015
STATUS
approved