A274260 Forbidden residues of the greedy prime offset sequence.

1, 1, 4, 3, 5, 1, 7, 9, 11, 25, 15, 33, 13, 21, 23, 31, 29, 52, 33, 35, 35, 39, 41, 58, 11, 13, 51, 53, 57, 29, 63, 65, 43, 69, 119, 75, 122, 81, 83, 112, 89, 4, 95, 94, 174, 99, 105, 111, 113, 123, 107, 119, 228, 125, 223, 131, 126, 135, 201, 29, 141, 193

Offset: 1

Author

R. Michael Perry, Jun 16 2016

Keywords

• nonn

Comments

The greedy prime offset sequence, A135311, is the closepacked integer sequence, starting with 0, such that for no prime p does the sequence form a complete system of residues modulo p. Instead, at least one residue must be missing for p, this is the (conjectured to be unique) "forbidden residue" for p. The first few terms of the greedy sequence are 0, 2, 6, 8, 12, 18. For the first three primes: 2, 3, 5, the forbidden residues are, respectively: 1, 1, 4. More generally, a(n) gives the forbidden residue for the n-th prime number. Every prime, it appears, has a unique forbidden residue, but this is unproven as far as I know. If this is true then every prime has an "exhaustion number" which is the number of terms of the greedy sequence needed to exhaust all the other residues and determine which one is forbidden; see A274261.

Note: I discovered the greedy sequence many years ago and did a writeup including discussion of forbidden residues and exhaustion numbers. See LINKS.

Links

• R. Michael Perry, a number sequence relating to the closepacking of primes

Crossrefs

Cf. A135311, A274261.

Programs

Mathematica

b[n_] := Module[{set = {}, m = 0, p, q, r}, p = Prime[n];
  While[Length[set] < p - 1, m++; q = Mod[g[m], p];
   If[FreeQ[set, q], set = Append[set, q]]];
  r = Complement[Range[0, p - 1], set][[1]];
  {n, p, r, m}]
(* b[n] returns a 4-element list: {n, Prime[n], forbidden_residue[n], exhaustion_number[n]}. g is the greedy sequence, see A135311 for Mathematica code, where a[n]=g[n].*)