A211204 a(1) = 2; for n > 1, a(n) > a(n-1) is the smallest prime for which the set {a(1), a(2), ..., a(n)} lacks at least one residue modulo every odd prime less than or equal to a(n).

2, 3, 5, 11, 17, 23, 41, 47, 53, 71, 83, 101, 107, 113, 131, 137, 167, 173, 191, 197, 233, 251, 257, 263, 311, 317, 347, 353, 401, 431, 443, 461, 467, 503, 521, 563, 593, 641, 647, 653, 677, 683, 701, 743, 761, 773, 797, 827, 857, 863, 881, 911, 941, 947, 971

Offset

1,1

Comments

By construction, for every odd prime p > 1, the sequence does not contain a full residue system modulo p. For n >= 4, all differences a(n) - a(n-1) are multiples of 6; otherwise said, a(n) == 5 (mod 6).

Conjecture: The sequence contains infinitely many "twins" with a(n)-a(n-1) = 6.

All terms greater than 3 are 2 mod 3, so the sequence does not contain a complete residue system mod 3; all terms are not 4 mod 5, so the sequence does not contain a complete residue system mod 5; since 7 is absent in the sequence, there is not a complete residue system mod 7.

By the Chinese remainder theorem and Dirichlet's theorem on arithmetic progressions, the sequence is infinite. - Dimiter Skordev, Apr 05 2022

Links

Dimiter Skordev, Table of n, a(n) for n = 1..1000 (first 92 terms from Vladimir Shevelev and Peter J. C. Moses)

Prog

(Python)
def isPrime(n):
    if (n%2==0): return n==2
    for i in range(3, int(n**0.5+1), 2):
        if (n%i==0): return False
    return n>1
def nextPrime(n):
    n=n+1
    while not isPrime(n): n=n+1
    return n
def a(n):
    p, L, S=2, [], []
    while len(L)<n-1:
        p, S1, i=nextPrime(p), S, 0
        while (i<len(L)) and ((len(S[i])+2<L[i]) or (p%L[i] in S[i])):
            S1[i].add(p%L[i])
            i=i+1
        if i==len(L):
            S1.append(set(L))
            S=S1
            L.append(p)
    return p
# Dimiter Skordev, Apr 05 2022

Crossrefs

Cf. A210537.

Sequence in context: A049595 A258039 A107438 * A023206 A359555 A049565

Adjacent sequences: A211201 A211202 A211203 * A211205 A211206 A211207

Keyword

nonn

Author

Vladimir Shevelev and Peter J. C. Moses, Feb 04 2013

Extensions

Edited by M. F. Hasler, Feb 13 2013

Status

approved