A135311 -id:A135311 - cp's OEIS frontend

A274261 Exhaustion numbers for the greedy prime offset sequence A135311.

1, 2, 4, 6, 11, 14, 19, 37, 38, 53, 50, 57, 80, 81, 99, 125, 131, 213, 156, 330, 161, 220, 173, 207, 244, 225, 312, 337, 234, 293, 462, 471, 434, 535, 349, 458, 470, 489, 477, 413, 527, 474, 619, 539, 554, 666, 743, 690, 1295, 740, 627, 617, 706, 911, 755, 867

Offset: 1

R. Michael Perry, Jun 16 2016

nonn

The greedy prime offset sequence, A135311, is the close-packed 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. Every prime, it appears, has a unique forbidden residue. 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. The uniqueness of the forbidden residue for any individual prime can be verified by calculation.

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

			The first few terms of the greedy offset sequence are 0, 2, 6, 8. For n=3, the n-th prime = 5. The residues of the greedy sequence modulo 5 are 0, 2, 1, 3 .... The first four residues exhaust all the possibilities but one, showing that 4 is the forbidden residue for 5 and the exhaustion number is also 4.

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

Cf. A135311, A274260.

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].*)

A213647 Initial members of prime 11-tuplets: primes p such that p + (0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36) are all prime.

Original entry on oeis.org

11, 7908189600581, 10527733922591, 12640876669691, 38545620633251, 43564522846961, 60268613366231, 60596839933361, 71431649320301, 79405799458871, 109319665100531, 153467532929981, 171316998238271, 216585060731771, 254583955361621, 259685796605351, 268349524548221

Offset: 1

Matt C. Anderson, Jun 17 2012

nonn

0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 are the first terms of A135311.
All terms are congruent to 11 (modulo 210). - Zak Seidov, Sep 15 2014
Subsequence of A202282. - Zak Seidov, Sep 15 2014
All terms, except the first one, are congruent to 1271 (modulo 2310). - Matt C. Anderson, May 29 2015

Matt C. Anderson and Dana Jacobsen, Table of n, a(n) for n = 1..6800 (first 100 terms from Matt C. Anderson)
Norman Luhn, Table of n, a(n) for n = 1..100000

Cf. A135311, A202282.

use ntheory ":all"; say for sieve_prime_cluster(1,1e14, 2,6,8,12,18,20,26,30,32,36); # Dana Jacobsen, Oct 01 2015

a(89) corrected by Dana Jacobsen, Oct 01 2015

A213645 Initial members of prime 12-tuplets. Primes p such that p + (0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42) are all prime.

Original entry on oeis.org

11, 380284918609481, 437163765888581, 701889794782061, 980125031081081, 1277156391416021, 1487854607298791, 1833994713165731, 2115067287743141, 2325810733931801, 3056805353932061, 3252606350489381, 3360877662097841, 3501482688249431, 3595802556731501

Offset: 1

Matt C. Anderson, Jun 17 2012

nonn

All terms, except the first one, are congruent to 1271 (modulo 2310). - Matt C. Anderson, May 29 2015

Matt C. Anderson and Dana Jacobsen, Table of n, a(n) for n = 1..2807 [first 83 entries by Matt C. Anderson]
Tony Forbes and Norman Luhn, prime k-tuplets
Norman Luhn, Table of n, a(n) for n = 1..20000
Wikipedia, Prime k-tuple

Cf. A022545, A022546, A022547, and A022548 (prime 9-tuplets).

Cf. A135311, 2*A101448 (both begin with 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42).

use ntheory ":all"; say for sieve_prime_cluster(1, 10**15, 2,6,8,12,18,20,26,30,32,36,42); # Dana Jacobsen, Oct 04 2015

Corrected and extended by Dana Jacobsen, Oct 04 2015

A274260 Forbidden residues of the greedy prime offset sequence.

Original entry on oeis.org

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

R. Michael Perry, Jun 16 2016

nonn

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.

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

Cf. A135311, A274261.

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].*)

A217552 A constant relating to generating primes from fractions involving Bernoulli numbers, and the greedy sequence of prime offsets.

Original entry on oeis.org

3, 3, 0, 5, 2, 1, 9, 8, 2, 9, 1, 8, 5, 3, 4, 4, 6, 8, 2, 1, 9, 1, 9, 8, 2, 8, 1, 1, 4, 7, 7, 2, 6, 2, 9, 2, 0, 9, 0, 9, 2, 2, 8, 4, 4, 8, 1, 2, 3, 3, 1, 3, 5, 2, 5, 8

Offset: 1

Roger Thompson, Oct 06 2012

If the prime k-tuple conjecture is true, then lim inf A217926(n)/(2n) exists, and has the value 3.305219...

If lim sup A217926(n)/(2n) exists, it appears to have a value less than 3.32.

Cf. A217926, A135311.

The a(n) are the decimal digits of K = 3.305219..., where K is the solution of exp(2/K) = sum(exp(-g/K)), and where the g are the terms of the greedy sequence of prime offsets (A135311).

A161781 Binary encodings of prime constellations.

Original entry on oeis.org

1, 3, 5, 9, 11, 13, 17, 19, 25, 27, 33, 37, 41, 45, 65, 67, 69, 73, 75, 77, 81, 83, 89, 91, 97, 101, 105, 109, 129, 131, 137, 139, 145, 147, 153, 193, 195, 201, 203, 209, 211, 257, 261, 265, 269, 289, 293, 297, 301, 321, 325, 329, 333, 353, 357, 361, 365, 513, 515

Offset: 0

Carl R. White, Jun 19 2009

nonn

Each constellation is encoded by means of dividing each of the increments to p in the k-tuple by two, raising two to the power of each and then summing the result. E.g.:
  (p,p+2,p+6) -> p+(0,2,6) => (0,1,3) -> 2^0 + 2^1 + 2^3 = 11.
Each encoding is unique and so can be reversed e.g.:
  89 = 2^0 + 2^3 + 2^4 + 2^6 -> (0,3,4,6) => (p,p+6,p+8,p+12).
Those constellations that represent all moduli for all their matching primes p are not counted; for example, encoding #7, which implies (p,p+2,p+4) only matches the prime triple (3,5,7) which is (0,2,1) mod 3, and so is not a valid constellation, and thus 7 is not in the list. Encoding #155 is the first that fails modulo 5, and is also not in the list.

			Encoding #1 corresponds to the primes themselves (constellations of one), #3 corresponds to the twin primes (p,p+2), #5 to the cousin primes (p,p+4) and #9 to the "sexy" primes (p,p+6).

Eric Weisstein's World of Mathematics, Prime Constellation.

Cf. A008407, A020497, A094660, A135311. Also compare A014657 which is unrelated but remarkably similar.

cp's OEIS Frontend

A274261 Exhaustion numbers for the greedy prime offset sequence A135311.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A213647 Initial members of prime 11-tuplets: primes p such that p + (0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36) are all prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Perl

Extensions

A213645 Initial members of prime 12-tuplets. Primes p such that p + (0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42) are all prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Perl

Extensions

A274260 Forbidden residues of the greedy prime offset sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A217552 A constant relating to generating primes from fractions involving Bernoulli numbers, and the greedy sequence of prime offsets.

Views

Author

Keywords

Comments

Crossrefs

Formula

A161781 Binary encodings of prime constellations.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs