A023048 -id:A023048 - cp's OEIS frontend

A002230 Primes with record values of the least positive primitive root.

2, 3, 7, 23, 41, 71, 191, 409, 2161, 5881, 36721, 55441, 71761, 110881, 760321, 5109721, 17551561, 29418841, 33358081, 45024841, 90441961, 184254841, 324013369, 831143041, 1685283601, 6064561441, 7111268641, 9470788801, 28725635761, 108709927561, 386681163961, 1990614824641, 44384069747161, 89637484042681

Offset: 1

N. J. A. Sloane

R. Osborn, Tables of All Primitive Roots of Odd Primes Less Than 1000, Univ. Texas Press, 1961.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.

Michel Marcus, Table of n, a(n) for n = 1..38 (using McGown and Sorenson).
Stephen D. Cohen, Tomás Oliveira e Silva, and Tim Trudgian, On Grosswald's conjecture on primitive roots, arXiv:1503.04519 [math.NT], 2015.
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Kevin J. McGown and Jonathan P. Sorenson, Computation of the least primitive root, arXiv:2206.14193 [math.NT], 2022.
Tomás Oliveira e Silva, Least prime primitive root of prime numbers
A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots, Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968 [Annotated scans of selected pages]
Index entries for primes by primitive root

Cf. A002229 (for the primitive roots in question).

Records in A023048, indices in A114885.

s = {2}; rm = 1; Do[p = Prime[k]; r = PrimitiveRoot[p]; If[r > rm, Print[p]; AppendTo[s, p]; rm = r], {k, 10^6}]; s (* Jean-François Alcover, Apr 05 2011 *)
DeleteDuplicates[Table[{p,PrimitiveRoot[p,1]},{p,Prime[Range[61100]]}],GreaterEqual[ #1[[2]],#2[[2]]]&][[All,1]] (* The program generates the first 15 terms of the sequence. *) (* Harvey P. Dale, Aug 22 2022 *)

from sympy import isprime, primitive_root
from itertools import count, islice
def f(n): return 0 if not isprime(n) or (r:=primitive_root(n))==None else r
def agen(r=0): yield from ((m, r:=f(m))[0] for m in count(1) if f(m) > r)
print(list(islice(agen(), 15))) # Michael S. Branicky, Feb 13 2023

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

A066529 a(n) is the least index such that the least primitive root of the a(n)-th prime is n, or zero if no such prime exists.

Original entry on oeis.org

1, 2, 4, 0, 9, 13, 20, 0, 0, 65, 117, 566, 88, 173, 85, 0, 64, 5426, 43, 10217, 80, 474, 326, 44110, 0, 1479, 0, 12443, 1842, 11662, 775, 0, 23559, 5029, 6461, 0, 3894, 5629, 15177, 105242, 14401, 182683, 9204, 7103, 5518399, 23888, 24092, 42304997, 0, 1455704, 27848, 12107, 14837, 205691645, 38451, 12102037, 39370, 28902, 57481, 56379, 90901, 53468, 5918705, 0, 732055, 1738826, 242495, 265666, 10523, 388487, 260680

Offset: 1

Wouter Meeussen, Jan 06 2002

nonn

The corresponding primes are in A023048.

For n < 150, only a(108) is presently unknown. - Robert G. Wilson v, Jan 03 2006

			a(6) = 13 because Prime[13] = 41 is the least prime with least primitive root = 6

Tomás Oliveira e Silva, Least prime primitive root of prime numbers
E. Weisstein, Primitive Roots
Index entries for primes by primitive root

Cf. A001918, A001122, A001123, A023048, A001597.

big = Table[ p = Prime[ n ]; PrimitiveRoot[ p ], {n, 1, 1000000} ]; Flatten[ Table[ Position[ big, n, 1, 1 ]/.{}-> 0, {n, 79} ] ] (* First load package NumberTheory`NumberTheoryFunctions` *)
(* first load package *) << NumberTheory`NumberTheoryFunctions` (* then do *) t = Table[0, {100}]; Do[a = PrimitiveRoot@Prime@n; If[a < 101 && t[[a]] == 0, t[[a]] = n], {n, 10^6}]; t (* Robert G. Wilson v, Dec 15 2005 *)

a(n) = 0 iff n is a perfect power (A001597) > 1. - Robert G. Wilson v, Jan 03 2006

a(n) = min { k | A001918(k) = n } U {0} = A000720(A023048(n)) (or zero). - M. F. Hasler, Jun 01 2018

Edited by Dean Hickerson, Jan 14 2002

Further terms from Robert G. Wilson v, Jan 03 2006

A133432 Let m = n-th number that is not a perfect power, A007916(n). Then a(n) = smallest prime having least positive primitive root m.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 29 2007

nonn

a(n) = A023048(A007916(n)).

N. J. A. Sloane, Table of n, a(n) for n=1..94 (from the web page of Tomás Oliveira e Silva)
Tomás Oliveira e Silva, Least primitive root of prime numbers

Cf. A023048, A007916, A001597, A133433 (records).

Definition corrected by Christopher J. Smyth, Oct 16 2013

A133433 Records in A133432.

Original entry on oeis.org

3, 7, 23, 41, 71, 313, 643, 4111, 53173, 107227, 533821, 1373989, 2494381, 95525767, 823766851, 4348468741, 226547941621, 596653488817, 5260410488191

Offset: 1

N. J. A. Sloane, Nov 29 2007

Also with an initial 2, records in A023048. The next term, which exceeds 10^14, is A023048(108) if it is not 0 (i.e., if there does exist a prime whose smallest primitive root is 108, then it is larger than 10^14). - Jianing Song, Aug 01 2018

Tomás Oliveira e Silva, Least primitive root of prime numbers

Cf. A023048, A133432.

A214158 Smallest number with n as least nonnegative primitive root, or 0 if no such number exists.

Original entry on oeis.org

1, 2, 3, 4, 0, 6, 41, 22, 0, 0, 313, 118, 4111, 457, 1031, 439, 0, 262, 53173, 191, 107227, 362, 3361, 842, 533821, 0, 12391, 0, 133321, 2906, 124153, 2042, 0, 3062, 48889, 2342, 0, 7754, 55441, 19322, 1373989, 3622, 2494381, 16022, 71761, 613034, 273001, 64682, 823766851, 0, 23126821, 115982, 129361, 29642

Offset: 0

Arkadiusz Wesolowski, Jul 05 2012

nonn

a(A001597(n)) = 0 for n > 1.

			a(7) = 22, since 22 has 7 as smallest positive primitive root and no number < 22 has 7 as smallest positive primitive root.

G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 313
Eric Weisstein's World of Mathematics, Primitive Root
Robert G. Wilson v, Table of n, a(n) for n = 0..205 (contains -1 where a term has not yet been found)

Cf. A046145, A046147, A023048.

lst2 = {}; r = 47; 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]; lst1 = Union[Flatten@Table[n^i, {i, 2, Log[2, r]}, {n, 2, r^(1/i)}]]; Do[n = 2; If[MemberQ[lst1, l], AppendTo[lst2, 0], While[True, If[smallestPrimitiveRoot[n] == l, AppendTo[lst2, n]; Break[]]; n++]], {l, r}]; Prepend[lst2, 1] (* Most of the code is from Jean-François Alcover *)

A083701 Smallest prime having Fibonacci(n) as least primitive root, or 0 if no such prime exists.

Original entry on oeis.org

2, 2, 3, 7, 23, 0, 457, 409, 48889, 459841, 6366361

Offset: 1

Sven Simon, May 04 2003

Next term >= 276000000. - Robert G. Wilson v, May 12 2003

			a(8) = 409 because 409 is the first prime having Fibonacci(8) = 21 as least primitive root

Cf. A023048, A083702.

<< NumberTheory`NumberTheoryFunctions`; a = Table[ Fibonacci[i], {i, 2, 20}]; b = Table[0, {20}]; k = 1; Do[k = NextPrime[k]; j = FromDigits[ Flatten[ Position[ a, PrimitiveRoot[k]]]]; If[ b[[j]] == 0, b[[j]] = k], {n, 1, 10^6}]; b

a(n) = A023048(A000045(n)). - Jianing Song, Oct 31 2018

Offset 1 from Michel Marcus, Oct 30 2018

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

A259484 Smallest nonprime number having least positive primitive root n, or 0 if no such root exists.

Original entry on oeis.org

1, 0, 9, 4, 0, 6, 1681, 22, 0, 0, 97969, 118, 16900321, 914, 1062961, 542, 0, 262, 2827367929, 382

Offset: 0

Robert G. Wilson v, Jun 28 2015

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.

			a(2) = 9 because the least primitive root of the nonprime number 9 is 2 and no nonprime less than 9 meets this criterion.

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.

Eric Weisstein's World of Mathematics, Primitive Root.

Cf. A001597, A023048, A046145, A214158.

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

a(n) = 0 if n is a perfect power (A001597).

a(18)-a(19) from Robert G. Wilson v, Sep 26 2015

A318239 a(n) is the smallest primitive root of A133433(n).

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 18, 20, 24, 40, 42, 45, 48, 54, 72, 80, 96

Offset: 1

Jianing Song, Aug 22 2018

Also with an initial 1, numbers k such that A023048(k) set a new record.
If A023048(108) != 0 (which is implied assuming generalized Artin's conjecture) then the next term of this sequence is 108.
Conjecturally 11 is the largest prime in this sequence, 42 is the largest squarefree term, and 45 is the largest odd term.

			The smallest prime with least primitive root 11 is 643, and the smallest prime with least primitive root 2, 3, 5, ..., 10 are all < 643, so 11 is a term.
The smallest prime with least primitive root 45 is 95525767, and the smallest prime with least primitive root 2, 3, 5, ..., 44 are all < 95525767, so 45 is a term.
13 is not a term since the smallest prime with least primitive root 13 is 457, but the smallest prime with least primitive root 12 is 4111, which is larger than 457.

Tomás Oliveira e Silva, Least primitive root of prime numbers

Cf. A023048, A133433.

cp's OEIS Frontend

A002230 Primes with record values of the least positive primitive root.

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A066529 a(n) is the least index such that the least primitive root of the a(n)-th prime is n, or zero if no such prime exists.

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A133432 Let m = n-th number that is not a perfect power, A007916(n). Then a(n) = smallest prime having least positive primitive root m.

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A133433 Records in A133432.

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A214158 Smallest number with n as least nonnegative primitive root, or 0 if no such number exists.

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Mathematica

A083701 Smallest prime having Fibonacci(n) as least primitive root, 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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A079061 Smallest prime p such that the least positive primitive root of p equals prime(n).

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