A193305 Composite numbers of the form 4, p^m, or 2*p^m for p an odd prime. All composites that have a primitive root.

4, 6, 9, 10, 14, 18, 22, 25, 26, 27, 34, 38, 46, 49, 50, 54, 58, 62, 74, 81, 82, 86, 94, 98, 106, 118, 121, 122, 125, 134, 142, 146, 158, 162, 166, 169, 178, 194, 202, 206, 214, 218, 226, 242, 243, 250, 254, 262, 274, 278, 289, 298, 302, 314, 326, 334, 338, 343

Offset

1,1

Comments

Nonprime k such that the multiplicative group modulo k is cyclic. Nonprime terms of A033948 (omitting the initial term 1). - Joerg Arndt, Aug 07 2011

a(n) has a primitive root for any n. - Arkadiusz Wesolowski, Sep 06 2012 [See, e.g., the Niven et al. reference. - Wolfdieter Lang, Jan 18 2017]

References

Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991, Theorem 2.41, p. 104.

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Joerg Arndt, Matters Computational (The Fxtbook), relation (39.7-13) on page 779.

Mathematica

lim = 500; t = {4}; Do[p = Prime[n]; k = 1; While[p^k <= lim, If[k > 1, AppendTo[t, p^k]]; If[2*p^k <= lim, AppendTo[t, 2*p^k]]; k++], {n, 2, PrimePi[lim/2]}]; Sort[t]; (* T. D. Noe, Sep 06 2012 *)

Prog

(PARI) for (n=2, 555, if ( isprime(n), next() ); if ( 1 == #(znstar(n)[3]), print1(n, ", ") ); );  /* Joerg Arndt, Aug 07 2011 */

Crossrefs

Cf. A033948, A033949 (composites without primitive root). A279398.

Sequence in context: A337372 A317299 A236026 * A084759 A054395 A142863

Adjacent sequences: A193302 A193303 A193304 * A193306 A193307 A193308

Keyword

nonn

Author

Warren Breslow, Jul 21 2011

Extensions

More terms from Joerg Arndt, Aug 07 2011

Name corrected and augmented by Wolfdieter Lang, Jan 18 2017

Status

approved