|
|
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
|
|
|
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
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|