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A259484 Smallest nonprime number having least positive primitive root n, or 0 if no such root exists. 0
1, 0, 9, 4, 0, 6, 1681, 22, 0, 0, 97969, 118, 16900321, 914, 1062961, 542, 0, 262, 2827367929, 382 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..19.

Eric Weisstein's World of Mathematics, Primitive Root.

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.

a(n) = 0 if n is a perfect power (A001597).

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-Fran├žois 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

Cf. A001597, A023048, A046145, A214158.

Sequence in context: A239349 A198675 A203131 * A021918 A112146 A056897

Adjacent sequences:  A259481 A259482 A259483 * A259485 A259486 A259487

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

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Last modified January 23 02:34 EST 2019. Contains 319365 sequences. (Running on oeis4.)