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
Keywords
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.
Programs
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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 *) -
PARI
for (n=2, 555, if ( isprime(n), next() ); if ( 1 == #(znstar(n)[3]), print1(n,", ") ); ); /* Joerg Arndt, Aug 07 2011 */
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Python
from sympy import primepi, integer_nthroot def A193305(n): def bisection(f,kmin=0,kmax=1): while f(kmax) > kmax: kmax <<= 1 kmin = kmax >> 1 while kmax-kmin > 1: kmid = kmax+kmin>>1 if f(kmid) <= kmid: kmax = kmid else: kmin = kmid return kmax def f(x): return int(n+x-(x>=4)-sum(primepi(integer_nthroot(x,k)[0])-1 for k in range(2,x.bit_length()))-sum(primepi(integer_nthroot(x>>1,k)[0])-1 for k in range(1,x.bit_length()-1))) return bisection(f,n,n) # Chai Wah Wu, Feb 24 2025
Extensions
More terms from Joerg Arndt, Aug 07 2011
Name corrected and augmented by Wolfdieter Lang, Jan 18 2017
Comments