A358238 a(n) is the least prime p such that the primes from prime(n) to p contain a complete set of residues modulo prime(n).

3, 7, 19, 29, 71, 103, 103, 191, 233, 317, 577, 439, 587, 467, 967, 659, 709, 1511, 1013, 1321, 1789, 1319, 1663, 2029, 1499, 2143, 1973, 2459, 2333, 2203, 3697, 3089, 3923, 4793, 3449, 4517, 3539, 4451, 3923, 4801, 5501, 4799, 4793, 7121, 5651, 4969, 6359, 4793, 6581, 9371, 6043, 9769, 5813

Offset

1,1

Comments

a(n) >= prime(n+prime(n)-1), with equality for n = 1, 2 and 4. Any others?

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Example

a(3) = 19 because prime(3) = 5 and the primes 5, 7, 11, 13, 17, 19 contain a complete set of residues mod 5: 5 == 0, 11 == 1, 7 and 17 == 2, 13 == 3, 19 == 4 (mod 5).

Maple

f:= proc(n) local p, S, q;
  p:= ithprime(n);
  S:= {$1..p-1};
  q:= p;
  do
    q:= nextprime(q);
    S:= S minus {q mod p};
    if S = {} then return q fi;
  od;
end proc:
map(f, [$1..100]);

Mathematica

a[n_] := Module[{p = Prime[n], s = {}, q}, q = p; While[Length[s = Union[AppendTo[s, Mod[q, p]]]] < p, q = NextPrime[q]]; q]; Array[a, 50] (* Amiram Eldar, Jan 31 2023 *)

Prog

(Python)
from sympy import prime, nextprime
def a(n):
    pn = p = prime(n); res = set()
    while True:
        res.add(p%pn)
        if len(res) == pn: return p
        p = nextprime(p)
print([a(n) for n in range(1, 54)]) # Michael S. Branicky, Jan 31 2023

Crossrefs

Cf. A360228.

Sequence in context: A091738 A340752 A360228 * A136054 A006032 A240072

Adjacent sequences: A358235 A358236 A358237 * A358239 A358240 A358241

Keyword

nonn

Author

Robert Israel, Jan 31 2023

Status

approved