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.

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

Offset: 2

Author

Charles R Greathouse IV, Sep 05 2011

Keywords

• nonn
• nice

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 that a(n) >> log(n)^2 * phi(n)/n.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 2..10000
• 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.
• 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.
• C. Pomerance, A note on the least prime in an arithmetic progression, Journal of Number Theory 12 (1980), pp. 218-223.

Crossrefs

Cf. A085420. Records are in A362850, A362851.

Programs

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-François Alcover, Oct 02 2013, translated from Pari *)

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

Python

from math import gcd
from gmpy2 import is_prime
def p(b, d):
    k = d + b
    while not is_prime(k):
        k += d
    return (k-b)//d
def A194943(n):
    return max(p(b, n) for b in range(1, n) if gcd(b, n) == 1)
print([A194943(n) for n in range(2, 82)]) # Michael S. Branicky, May 18 2023 after Charles R Greathouse IV

Formula

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