1,2

Given a(i) for 1 <= i < n, a(n) is the smallest number > a(n-1) such that, for every prime p, the set {a(i) mod p : 1<=i<=n} has at most p-1 elements. Assuming Schinzel's hypothesis H, an equivalent statement is that a(n) is minimal such that there are infinitely many primes p with p+a(i) prime for 1 <= i <= n.

For every n, a(n) is not congruent to 1 (mod 2), nor to 1 (mod 3), nor to 4 (mod 5), nor to 3 (mod 7), ...

Note that this sequence does not always give the minimal difference between the first and last of n consecutive large primes, A008407. E.g. a(6)=18 but the 6 consecutive primes 97, 101, 103, 107, 109, 113 give the minimal difference of 16.

Thomas J. Engelsma, K-Tuple Permissible Patterns.

Anthony D. Forbes, Prime k-tuplets.

MathWorld, Prime Constellation.

Wikipedia, Schinzel's hypothesis H.

Given a(1) through a(5), a(6) can't be 14 since the set {0,2,6,8,12,14} contains elements from every residue class (mod 5). a(6) can't be 16 because {0,2,6,8,12,16} contains elements from every residue class (mod 3). a(6)=18 is possible, since the residues (mod 2) are all 0, the residues (mod 3) are all 0 or 2 and the residues (mod 5) are all 0, 1, 2, or 3.

a[1]=0; a[n_]:=a[n]=Module[{v, set, ok, p}, For[v=a[n-1]+2, True, v+=2, set=Append[a/@Range[n-1], v]; For[p=3; ok=True, p<=n, p+=2, If[PrimeQ[p]&&Length[Union[Mod[set, p]]]==p, ok=False; Break[]]]; If[ok, Return[v]]]]

Cf. A008407.

Sequence in context: A008407 A111224 A139718 this_sequence A140115 A162860 A093006

Adjacent sequences: A135308 A135309 A135310 this_sequence A135312 A135313 A135314

nonn

galathaea(AT)gmail.com, Dec 07 2007

Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Dec 07 2007