

A274260


Forbidden residues of the greedy prime offset sequence.


2



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


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 nth 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

Table of n, a(n) for n=1..62.
R. Michael Perry, a number sequence relating to the closepacking of primes


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 4element 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].*)


CROSSREFS

Cf. A135311, A274261.
Sequence in context: A316254 A029934 A246665 * A011397 A081665 A131911
Adjacent sequences: A274257 A274258 A274259 * A274261 A274262 A274263


KEYWORD

nonn


AUTHOR

R. Michael Perry, Jun 16 2016


STATUS

approved



