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

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

Author

Benoit Cloitre, Feb 02 2003

Keywords

• nonn

Comments

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

Links

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

Crossrefs

Cf. A001918, A023048, A066529.

Programs

PARI

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

PARI

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

Python

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

Python

# 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

Extensions

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

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