A135311 A greedy sequence of prime offsets.

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

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