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A194943 Greatest d such that d*n+b is the least prime in the arithmetic progression k*n+b for some 0 < b < n with gcd(b, n) = 1. 3
1, 2, 1, 3, 1, 4, 2, 4, 1, 6, 1, 7, 3, 2, 2, 6, 2, 10, 2, 4, 3, 10, 3, 10, 3, 6, 2, 10, 2, 18, 4, 6, 5, 6, 4, 11, 5, 5, 3, 14, 2, 10, 5, 8, 6, 20, 3, 12, 5, 8, 11, 12, 3, 6, 4, 7, 5, 12, 2, 24, 9, 6, 5, 6, 3, 15, 5, 8, 3, 18, 4, 24, 8, 8, 6, 10 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

a(n) exists due to Linnik's theorem; thus a(n) < c * n^4.2 for some constant c.

Heath-Brown's conjecture on Linnik's theorem implies that a(n) < n.

On the GRH, a(n) << phi(n) * log(n)^2 * phi(n)/n.

Pomerance shows that a(n) > (e^gamma + o(1)) log(n) * phi(n)/n, and Granville & Pomerance conjecture than a(n) >> log(n)^2 * phi(n)/n.

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 2..10000

D. R. Heath-Brown, Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression, Proceedings of the London Mathematical Society 3:64 (1992), pp. 265-338.

A. Granville and C. Pomerance, On the least prime in certain arithmetic progressions, Journal of the London Mathematical Society 2:41 (1990), pp. 193-200.

C. Pomerance, A note on the least prime in an arithmetic progression, Journal of Number Theory 12 (1980), pp. 218-223.

FORMULA

a(n) = floor(A085420(n)/n).

MATHEMATICA

p[b_, d_] := Module[{k = b+d}, While[ !PrimeQ[k], k += d]; (k-b)/d]; a[n_] := Module[{r = p[1, n]}, For[b = 2, b <= n-1, b++, If[GCD[b, n] > 1, Null,  r = Max[r, p[b, n]]]]; r]; Table[a[n], {n, 2, 100}] (* Jean-Fran├žois Alcover, Oct 02 2013, translated from Pari *)

PROG

(PARI) p(b, d)=my(k=d+b); while(!isprime(k), k+=d); (k-b)/d

a(n)=my(r=p(1, n)); for(b=2, n-1, if(gcd(b, n)>1, next); r=max(r, p(b, n))); r

CROSSREFS

Cf. A085420, A034693. Records are in A194944, A194945.

Sequence in context: A325828 A200780 A338899 * A087145 A117172 A029207

Adjacent sequences:  A194940 A194941 A194942 * A194944 A194945 A194946

KEYWORD

nonn,nice

AUTHOR

Charles R Greathouse IV, Sep 05 2011

STATUS

approved

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Last modified July 31 02:33 EDT 2021. Contains 346367 sequences. (Running on oeis4.)