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A135311
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A greedy sequence of prime offsets.
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1
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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; internal format)
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OFFSET
| 1,2
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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.
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REFERENCES
| 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, 2011
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LINKS
| Thomas J. Engelsma, K-Tuple Permissible Patterns.
Anthony D. Forbes, Prime k-tuplets.
Eric Weisstein's World of Mathematics, Prime Constellation
Wikipedia, Schinzel's hypothesis H.
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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.
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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]]]]
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CROSSREFS
| Cf. A008407.
Sequence in context: A111224 A139718 A173340 * A200568 A140115 A162860
Adjacent sequences: A135308 A135309 A135310 * A135312 A135313 A135314
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KEYWORD
| nonn
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AUTHOR
| galathaea(AT)gmail.com, Dec 07 2007
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EXTENSIONS
| Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Dec 07 2007
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