

A034693


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


50



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
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OFFSET

1,3


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
This is true by Dirichlet's theorem on arithmetic progressions: for any pair (r,s) of integers such that gcd(r,s)=1 there are infinitely many primes in the sequence r + k*s; choose r=1 and s=p.  Joerg Arndt, Mar 20 2016
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
It is known that, for almost all n, a(n) <= n^2. From HeathBrown'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


REFERENCES

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 127130.
Ribenboim, P. (1989), The Book of Prime Number Records. Chapter 4, Section IV.B.: The Smallest Prime In Arithmetic Progressions, pp. 217223.


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000
S. R. Finch, More about Linnik's Constant
D. Graham, On Linnik's Constant, Acta Arithm., 39, 1981, pp. 163179.
D. R. HeathBrown, Zerofree regions for Dirichlet Lfunctions, and the least prime in an arithmetic progression, Proc. London Math. Soc. 64(3) (1992), pp. 265338.
I. Niven and B. Powell, Primes in Certain Arithmetic Progressions, Amer. Math. Monthly, 83, 1976, pp. 467489.
Wikipedia, Dirichlet's theorem on arithmetic progressions


FORMULA

It seems that Sum_{k=1..n} a(k) is asymptotic to (zeta(2)1)*n*log(n) where zeta(2)1 = Pi^2/61 = 0.6449... .  Benoit Cloitre, Aug 11 2002


EXAMPLE

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


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}] (* JeanFrançois Alcover, Jul 19 2011 *)


PROG

(PARI) a(n)=if(n<0, 0, s=1; while(isprime(s*n+1)==0, s++); s)
(Haskell)
a034693 n = head [k  k < [1..], a010051 (k * n + 1) == 1]
 Reinhard Zumkeller, Feb 14 2013


CROSSREFS

Cf. A010051, A034694, A053989, A071558, A085420, A103689, A194945, A200996.
Sequence in context: A205403 A080825 A229724 * A216506 A072342 A257089
Adjacent sequences: A034690 A034691 A034692 * A034694 A034695 A034696


KEYWORD

nonn,nice


AUTHOR

Labos Elemer


STATUS

approved



