A034693 Smallest k such that k*n+1 is prime.

1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 2, 1, 4, 2, 2, 1, 6, 1, 10, 2, 2, 1, 2, 3, 4, 2, 4, 1, 2, 1, 10, 3, 2, 3, 2, 1, 4, 5, 2, 1, 2, 1, 4, 2, 4, 1, 6, 2, 4, 2, 2, 1, 2, 2, 6, 2, 4, 1, 12, 1, 6, 5, 2, 3, 2, 1, 4, 2, 2, 1, 8, 1, 4, 2, 2, 3, 6, 1, 4, 3, 2, 1, 2, 4, 12, 2, 4, 1, 2, 2, 6, 3, 4, 3, 2, 1, 4, 2

Offset: 1

Author

Labos Elemer

Keywords

• nonn
• nice

Comments

Conjecture: for every n > 1 there exists a number k < n such that n*k + 1 is a prime. - Amarnath Murthy, Apr 17 2001

A stronger conjecture: for every n there exists a number k < 1 + n^(.75) such that n*k + 1 is a prime. I have verified this up to n = 10^6. Also, the expression 1 + n^(.74) does not work as an upper bound (counterexample: n = 19). - Joseph L. Pe, Jul 16 2002

Stronger version of the conjecture verified up to 10^9. - Mauro Fiorentini, Jul 23 2023

It is known that, for almost all n, a(n) <= n^2. From Heath-Brown's result (1992) obtained with help of the GRH, it follows that a(n) <= (phi(n)*log(n))^2. - Vladimir Shevelev, Apr 30 2012

Conjecture: a(n) = O(log(n)*log(log(n))). - Thomas Ordowski, Oct 17 2014

I conjecture the opposite: a(n) / (log n log log n) is unbounded. See A194945 for records in this sequence. - Charles R Greathouse IV, Mar 21 2016

Examples

			If n=7, the smallest prime in the sequence 8, 15, 22, 29, ... is 29, so a(7)=4.
		

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, section 2.12, pp. 127-130.
• P. Ribenboim, (1989), The Book of Prime Number Records. Chapter 4, Section IV.B.: The Smallest Prime In Arithmetic Progressions, pp. 217-223.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Gunther Cornelissen, David Hokken, and Berend Ringeling, The asymptotic Mahler measure of Gaussian periods, arXiv:2507.09303 [math.NT], 2025. See p. 1.
• Steven R. Finch, Linnik's Constant
• D. Graham, On Linnik's Constant, Acta Arithm., 39, 1981, pp. 163-179.
• D. R. Heath-Brown, Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. 64(3) (1992), pp. 265-338.
• Pengcheng Niu and Junli Zhang, On Two Conjectures of A. Murthy, ResearchGate (2024).
• Ivan Niven and Barry Powell, Primes in Certain Arithmetic Progressions, Amer. Math. Monthly, 83, 1976, pp. 467-489.
• Wikipedia, Dirichlet's theorem on arithmetic progressions

Crossrefs

Cf. A010051, A034694, A053989, A071558, A085420, A103689, A194944 (records), A194945 (positions of records), A200996.

Programs

Haskell

a034693 n = head [k | k <- [1..], a010051 (k * n + 1) == 1]
-- Reinhard Zumkeller, Feb 14 2013

Maple

A034693 := proc(n)
    for k from 1 do
        if isprime(k*n+1) then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Jul 26 2015

Mathematica

a[n_]:=(k=0; While[!PrimeQ[++k*n + 1]]; k); Table[a[n], {n,100}] (* Jean-François Alcover, Jul 19 2011 *)

PARI

a(n)=if(n<0,0,s=1; while(isprime(s*n+1)==0,s++); s)

Python

from sympy import isprime
def a(n):
    k = 1
    while not isprime(k*n+1): k += 1
    return k
print([a(n) for n in range(1, 99)]) # Michael S. Branicky, May 05 2022

Formula

It seems that Sum_{k=1..n} a(k) is asymptotic to (zeta(2)-1)*n*log(n) where zeta(2)-1 = Pi^2/6-1 = 0.6449... . - Benoit Cloitre, Aug 11 2002

a(n) = (A034694(n)-1) / n. - Joerg Arndt, Oct 18 2020