A210537 -id:A210537 - cp's OEIS frontend

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

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

nonn

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

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

Cf. A210537.

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)Dimiter Skordev, Apr 05 2022

Edited by M. F. Hasler, Feb 13 2013

A220533 a(n) is minimal number such that the set of all composite numbers <= a(n) contains complete residue system modulo n.

Original entry on oeis.org

4, 9, 8, 15, 12, 35, 14, 27, 20, 33, 24, 65, 26, 45, 32, 45, 36, 77, 38, 63, 44, 63, 46, 95, 48, 69, 56, 87, 60, 187, 62, 93, 64, 105, 72, 175, 74, 117, 80, 123, 84, 215, 86, 117, 92, 135, 94, 245, 96, 153, 104, 141, 106, 245, 108, 165, 116

Offset: 1

Vladimir Shevelev, Feb 20 2013

nonn

Cf. A038026, A210537, A211204.

A220533 := proc(n)
    local sco,c,ai,scor ;
    sco := {} ;
    for ai from 1 do
        sco := sco union {A002808(ai)} ;
        scor := convert( [seq(c mod n,c=sco)], set) ;
        if nops(scor) = n then
            return A002808(ai) ;
        end if;
    end do:
end proc:
seq(A220533(n),n=1..60) ; # R. J. Mathar, Feb 27 2013

A002808[n_] := A002808[n] = Module[{a}, If[ n == 1 , 4, For[a = A002808[n-1] + 1 , True, a++, If[! PrimeQ[a], Return [a]]]]]; A220533[n_] := Module[{ sco, c, ai, scor}, sco = {}; For[ai = 1, True, ai++, AppendTo[sco, A002808[ai]] ; scor = Mod[#, n]& /@ sco // Union; If[Length[scor] == n , Return[A002808[ai]]]]]; Table[A220533[n], {n, 1, 57}] (* Jean-François Alcover, Feb 28 2013, translated from R. J. Mathar's Maple program *)

For odd n, a(n) <= 2n + 2; the equality holds if and only if n + 2 is prime.

a(n)<2*n for 25,33,49,... - R. J. Mathar and Vladimir Shevelev, Feb 28 2013

Corrected and extended by R. J. Mathar, Feb 27 2013

cp's OEIS Frontend

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).

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