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A135311 A greedy sequence of prime offsets. 7
0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56, 62, 68, 72, 78, 86, 90, 96, 98, 102, 110, 116, 120, 128, 132, 138, 140, 146, 152, 156, 158, 162, 168, 176, 182, 186, 188, 198, 200, 210, 212, 216, 230, 240, 242, 246, 252, 260, 266, 270, 272, 278, 282 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

LINKS

Alessandro Languasco, Table of n, a(n) for n = 1..2089

Thomas J. Engelsma, K-Tuple Permissible Patterns.

Anthony D. Forbes, Prime k-tuplets.

Kevin Ford, Florian Luca and Pieter Moree, Values of the Euler phi-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields, arXiv preprint arXiv:1108.3805 [math.NT], 2011.

Alessandro Languasco, Efficient computation of the Euler-Kronecker constants of prime cyclotomic fields, Research in Number Theory, 7, 2021, paper n. 2 (preliminary version, arXiv:1903.05487 [math.NT], 2019-2020).

Alessandro Languasco, Pieter Moree, Sumaia Saad Eddin, and Alisa Sedunova, Computation of the Kummer ratio of the class number for prime cyclotomic fields, arXiv:1908.01152 [math.NT], 2019.

Pieter Moree, Irregular behaviour of class numbers and Euler-Kronecker constants of cyclotomic fields: the log log log devil at play, arXiv:1711.07996 [math.NT], 2017. Mentions this sequence.

Eric Weisstein's World of Mathematics, Prime Constellation

Wikipedia, Schinzel's hypothesis H.

EXAMPLE

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.

MATHEMATICA

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]]]]

PROG

(PARI)  {greedy()=local(A, L, B, n, v , ok , R, setR, p, k);

A=vector(2089); \\ 2089 is the length to get Sum_{i>=2}(1/A[i])>2; see Ford, Luca, Moree paper, p. 1454

L=length(A); B = 10^(5); \\ upper bound for the number of primes used; enough for the first 2089 terms

A[1]=0;  \\ first trivial term;

for (n=2, L,

R=vector(n);

forstep (v=A[n-1]+2, B, 2 , ok=1;

forprime(p = 2, v,

for(k=1, n-1, R[k]=A[k]%p);

R[n]=v%p;

setR=Set(R);

if (length(setR) > p-1, ok=0; break);  \\ v is not good

);

if (ok==1, A[n]=v; break);

);

);

return(A)

} \\ Alessandro Languasco, Aug 11 2019

CROSSREFS

Cf. A008407, A020497.

Sequence in context: A111224 A139718 A173340 * A200568 A162860 A093006

Adjacent sequences:  A135308 A135309 A135310 * A135312 A135313 A135314

KEYWORD

nonn

AUTHOR

galathaea(AT)gmail.com, Dec 07 2007

EXTENSIONS

Edited by Dean Hickerson, Dec 07 2007

STATUS

approved

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Last modified April 21 08:49 EDT 2021. Contains 343148 sequences. (Running on oeis4.)