A023048 - OEIS

%I #72 Feb 13 2023 11:16:07

%S 2,3,7,0,23,41,71,0,0,313,643,4111,457,1031,439,0,311,53173,191,

%T 107227,409,3361,2161,533821,0,12391,0,133321,15791,124153,5881,0,

%U 268969,48889,64609,0,36721,55441,166031,1373989,156601,2494381,95471,71761,95525767

%N Smallest prime having least positive primitive root n, or 0 if no such prime exists.

%C a(n) = 0 iff n is a perfect power m^k, m >= 1, k >= 2 (i.e., a member of A001597).

%C Of course if n is a perfect power then a(n) = 0, but it seems that the other direction is true only assuming the generalized Artin's conjecture. See the link from Tomás Oliveira e Silva below. - _Jianing Song_, Jan 22 2019

%D 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.

%H N. J. A. Sloane, <a href="/A023048/b023048.txt">Table of n, a(n) for n = 1..107</a> (from the web page of Tomás Oliveira e Silva)

%H Wouter Meeussen, <a href="/A023048/a023048_1.txt">Smallest Primes with Specified Least Primitive Root</a>

%H Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/p_roots.html">Least primitive root of prime numbers</a>

%H <a href="/index/Pri#primes_root">Index entries for primes by primitive root</a>

%F a(n) = min { prime(k) | A001918(k) = n } U {0} = A000040(A066529(n)) (or zero). - _M. F. Hasler_, Jun 01 2018

%e a(2) = 3, since 3 has 2 as smallest positive primitive root and no prime p < 3 has 2 as smallest positive primitive root.

%e a(24) = 533821, since prime 533821 has 24 as smallest positive primitive root and no prime p < 533821 has 24 as smallest positive primitive root.

%t t = Table[0, {100}]; Do[a = PrimitiveRoot@Prime@n; If[a < 101 && t[[a]] == 0, t[[a]] = n], {n, 10^6}]; Unprotect[Prime]; Prime[0] = 0; Prime@t; Clear[Prime]; Protect[Prime] (* _Robert G. Wilson v_, Dec 15 2005 *)

%o (Python)

%o from sympy import nextprime, perfect_power, primitive_root

%o def a(n):

%o     if perfect_power(n): return 0

%o     p = 2

%o     while primitive_root(p) != n: p = nextprime(p)

%o     return p

%o print([a(n) for n in range(1, 40)]) # _Michael S. Branicky_, Feb 13 2023

%o (Python) # faster version for initial segment of sequence

%o from itertools import count, islice

%o from sympy import nextprime, perfect_power, primitive_root

%o def agen(): # generator of terms

%o     p, adict, n = 2, {None: 0}, 1

%o     for k in count(1):

%o         v = primitive_root(p)

%o         if v not in adict:

%o             adict[v] = p

%o         if perfect_power(n): adict[n] = 0

%o         while n in adict: yield adict[n]; n += 1

%o         p = nextprime(p)

%o print(list(islice(agen(), 40))) # _Michael S. Branicky_, Feb 13 2023

%Y Cf. A001122-A001126, A061323-A061335, A061730-A061741.

%Y Indices of the primes: A066529.

%Y For records see A133433. See A133432 for a version without the 0's.

%K nonn

%O 1,1

%A _David W. Wilson_

%E Comment corrected by _Christopher J. Smyth_, Oct 16 2013