A266909 Table read by rows: for each k < n and coprime to n, the least x>=0 such that x*n+k is prime.

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

Offset: 1

Author

Robert Israel, Jan 05 2016

Keywords

• nonn
• tabf

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.

Examples

			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
		

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

Crossrefs

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

Programs

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 *)