A066529 -id:A066529 - cp's OEIS frontend

A023048 Smallest prime having least positive primitive root n, or 0 if no such prime exists.

2, 3, 7, 0, 23, 41, 71, 0, 0, 313, 643, 4111, 457, 1031, 439, 0, 311, 53173, 191, 107227, 409, 3361, 2161, 533821, 0, 12391, 0, 133321, 15791, 124153, 5881, 0, 268969, 48889, 64609, 0, 36721, 55441, 166031, 1373989, 156601, 2494381, 95471, 71761, 95525767

Offset: 1

David W. Wilson

nonn

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

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

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

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.

N. J. A. Sloane, Table of n, a(n) for n = 1..107 (from the web page of Tomás Oliveira e Silva)
Wouter Meeussen, Smallest Primes with Specified Least Primitive Root
Tomás Oliveira e Silva, Least primitive root of prime numbers
Index entries for primes by primitive root

Cf. A001122-A001126, A061323-A061335, A061730-A061741.
Indices of the primes: A066529.
For records see A133433. See A133432 for a version without the 0's.

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 *)

from sympy import nextprime, perfect_power, primitive_root
def a(n):
    if perfect_power(n): return 0
    p = 2
    while primitive_root(p) != n: p = nextprime(p)
    return p
print([a(n) for n in range(1, 40)]) # Michael S. Branicky, Feb 13 2023

# faster version for initial segment of sequence
from itertools import count, islice
from sympy import nextprime, perfect_power, primitive_root
def agen(): # generator of terms
    p, adict, n = 2, {None: 0}, 1
    for k in count(1):
        v = primitive_root(p)
        if v not in adict:
            adict[v] = p
        if perfect_power(n): adict[n] = 0
        while n in adict: yield adict[n]; n += 1
        p = nextprime(p)
print(list(islice(agen(), 40))) # Michael S. Branicky, Feb 13 2023

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

Comment corrected by Christopher J. Smyth, Oct 16 2013

A066814 Smallest prime p such that (p-1) has n divisors, or 0 if no such prime exists.

Original entry on oeis.org

2, 3, 5, 7, 17, 13, 0, 31, 37, 113, 0, 61, 0, 193, 401, 211, 65537, 181, 0, 241, 577, 13313, 0, 421, 1297, 12289, 4357, 2113, 0, 1009, 0, 1321, 25601, 2424833, 752734097, 1801, 0, 786433, 495617, 2161, 0, 4801, 0, 15361, 7057, 155189249, 0

Offset: 1

Wouter Meeussen, Jan 20 2002

nonn

The only primes p for which p-1 has a prime number of divisors are Fermat primes A019434.

			a(17)=65537 because DivisorSigma[0,65536]=17.

Cf. A004249, A007516, A066529.

it=Table[ p=Prime[ n ]; DivisorSigma[ 0, p-1 ], {n, 400000} ]; Flatten[ Position[ it, #, 1, 1 ]&/@Range[ 100 ]/.{}- > 0 ]

Comment clarified by T. D. Noe, Nov 06 2009

Edited by Max Alekseyev, Nov 10 2009

A079060 Least k such that the least positive primitive root of prime(k) equals prime(n).

Original entry on oeis.org

2, 4, 9, 20, 117, 88, 64, 43, 326, 1842, 775, 3894, 14401, 9204, 24092, 14837, 57481, 90901, 242495, 260680, 61005, 508929, 1084588, 436307, 1124509, 1824015, 2969632, 2052357, 4006960, 5241202, 10253662, 30802809, 17480124, 73915355, 98931475, 42664033

Offset: 1

Benoit Cloitre, Feb 02 2003

nonn

a(49) = 1247136427. For n > 45, a(n) > 1.5*10^9 except n = 49. - David A. Corneth, Feb 15 2023

David A. Corneth, Table of n, a(n) for n = 1..45

Cf. A001918, A023048, A066529.

a(n) = {my(p=prime(n), s=1); while(p!=lift(znprimroot(prime(s))), s++); s; } \\ Modified by Jinyuan Wang, Apr 03 2020

upto(u, {maxn = 100}) = { my(t = 1, m = Map(), res = []); forprime(p = 2, oo, mapput(m, p, t); t++; if(t > maxn, break ) ); t = 1; u = prime(u); forprime(p = 2, u, c = lift(znprimroot(p)); if(mapisdefined(m, c), ind = mapget(m, c); if(ind > #res, res = concat(res, vector(ind - #res)) ); if(res[ind] == 0, res[ind] = t; ) ); t++ ); res } \\ David A. Corneth, Feb 15 2023

from sympy import nextprime, primitive_root
def a(n):
    k, pk, pn = 1, 2, prime(n)
    while primitive_root(pk) != pn: k += 1; pk = nextprime(pk)
    return k
print([a(n) for n in range(1, 19)]) # Michael S. Branicky, Feb 13 2023

# faster version for segments of sequence
from itertools import count, islice
from sympy import isprime, nextprime, prime, primepi, primitive_root
def agen(startk=1, startn=1): # generator of terms
    p, vdict, adict, n = prime(startk), dict(), dict(), startn
    for k in count(startk):
        v = primitive_root(p)
        if v not in vdict and isprime(v):
            vdict[v] = k
            adict[primepi(v)] = k
        while n in adict: yield adict[n]; n += 1
        p = nextprime(p)
print(list(islice(agen(), 18))) # Michael S. Branicky, Feb 14 2023

a(17)-a(18) from Jinyuan Wang, Apr 03 2020

a(19)-a(36) from Michael S. Branicky, Feb 14 2023

A079061 Smallest prime p such that the least positive primitive root of p equals prime(n).

Original entry on oeis.org

3, 7, 23, 71, 643, 457, 311, 191, 2161, 15791, 5881, 36721, 156601, 95471, 275641, 161831, 712321, 1171921, 3384481, 3659401, 760321, 7510801, 16889161, 6366361, 17551561, 29418841, 49443241, 33358081, 67992961, 90441961, 184254841

Offset: 1

Benoit Cloitre, Feb 02 2003

nonn

Smallest prime(m) such that A001918(m) = prime(n). (Corrected by Jonathan Sondow, Feb 03 2013)

a(36) = 831143041 and a(34) and a(35) > 1065000000. - Robert G. Wilson v, Jul 03 2003

Cf. A001918, A023048, A066529, A084739.

<< NumberTheory`NumberTheoryFunctions`; a = Table[ 0, {36}]; p = 2; Do[p = NextPrime[p]; pr = PrimitiveRoot[p]; If[ PrimeQ[pr] && PrimePi[pr] < 37 && a[[ PrimePi[pr]]] == 0, a[[ PrimePi[ pr]]] = p], {n, 2, 54000000}]; a

a(n)=if(n<0,0,s=1; while(prime(n)!=lift(znprimroot(prime(s))),s++); prime(s))

a(n) = A023048(prime(n)). - R. J. Mathar, Aug 03 2018

More terms from Robert G. Wilson v, Jul 03 2003

A114885 Prime indices with record values of the least positive primitive root.

Original entry on oeis.org

1, 2, 4, 9, 13, 20, 43, 80, 326, 775, 3894, 5629, 7103, 10523, 61005, 355588, 1124509, 1824015, 2052357, 2719588, 5241202, 10253662, 17480124, 42664033, 83470664, 282411553, 328711697, 432040721, 1247136427, 4461350728, 15082139743

Offset: 1

Robert G. Wilson v, Dec 29 2005

nonn

Eric Weisstein's World of Mathematics, Primitive Root.

Cf. A002229 (for the primitive roots in question), the records themselves are in A066529.

cp's OEIS Frontend

A023048 Smallest prime having least positive primitive root n, or 0 if no such prime exists.

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A066814 Smallest prime p such that (p-1) has n divisors, or 0 if no such prime exists.

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A079060 Least k such that the least positive primitive root of prime(k) equals prime(n).

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PARI

Python

Python

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A079061 Smallest prime p such that the least positive primitive root of p equals prime(n).

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Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A114885 Prime indices with record values of the least positive primitive root.

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Author

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