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

Author

galathaea(AT)gmail.com, Dec 07 2007

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A008407, A020497.

Programs

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]]]]

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

Extensions

Edited by Dean Hickerson, Dec 07 2007