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A266909 Table read by rows: for each k < n and coprime to n, the least x>=0 such that x*n+k is prime. 3
1, 2, 0, 1, 0, 2, 0, 0, 3, 1, 0, 4, 0, 0, 1, 0, 1, 2, 0, 0, 0, 2, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 3, 0, 1, 0, 1, 2, 3, 1, 0, 0, 0, 4, 0, 0, 1, 0, 1, 0, 3, 4, 1, 0, 7, 2, 0, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 1, 6, 0, 0, 5, 0, 1, 0, 3, 2, 3, 0, 1, 0, 1, 4, 3, 1, 0, 0, 0, 0, 0, 10, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

By Dirichlet's theorem, such x exists whenever k is coprime to n.

By Linnik's theorem, there exist constants b and c such that T(n,k) <= b n^c for all n and all k < n coprime to n.

T(n,1) = A034693(n).

T(n,n-1) = A053989(n)-1.

T(prime(n),1) = A035096(n).

T(2^n,1) = A035050(n).

A085427(n) = T(2^n,2^n-1) + 1.

A126717(n) = 2*T(2^(n+1),2^n-1) + 1.

A257378(n) = 2*T(n*2^(n+1),n*2^n+1) + 1.

A257379(n) = 2*T(n*2^(n+1),n*2^n-1) + 1.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10975 (rows 2 to 190, flattened)

Wikipedia, Dirichlet's theorem on arithmetic progressions.

Wikipedia, Linnik's theorem

Index entries for sequences related to primes in arithmetic progressions

EXAMPLE

The first few rows are

n=2: 1

n=3: 2, 0

n=4: 1, 0

n=5: 2, 0, 0, 3

n=6: 1, 0

MAPLE

T:= proc(n, k) local x;

    if igcd(n, k) <> 1 then return NULL fi;

    for x from 0 do if isprime(x*n+k) then return x fi

    od

end proc:

seq(seq(T(n, k), k=1..n-1), n=2..30);

MATHEMATICA

Table[Map[Catch@ Do[x = 0; While[! PrimeQ[x n + #], x++]; Throw@ x, {10^3}] &, Range@ n /. k_ /; GCD[k, n] > 1 -> Nothing], {n, 2, 19}] // Flatten (* Michael De Vlieger, Jan 06 2016 *)

CROSSREFS

Cf. A034693, A035050, A035096, A053989, A085427, A126717, A257378, A257379.

Sequence in context: A089734 A321375 A217564 * A276491 A035177 A194591

Adjacent sequences:  A266906 A266907 A266908 * A266910 A266911 A266912

KEYWORD

nonn,tabf

AUTHOR

Robert Israel, Jan 05 2016

STATUS

approved

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Last modified March 18 16:00 EDT 2019. Contains 321292 sequences. (Running on oeis4.)