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A034693
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Smallest k such that k*n+1 is prime.
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51
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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
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1,3
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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If n=7, the smallest prime in the sequence 8, 15, 22, 29, ... is 29, so a(7)=4.
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MAPLE
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for k from 1 do
if isprime(k*n+1) then
return k;
end if;
end do:
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MATHEMATICA
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a[n_]:=(k=0; While[!PrimeQ[++k*n + 1]]; k); Table[a[n], {n, 100}] (* Jean-François Alcover, Jul 19 2011 *)
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PROG
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(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]
(Python)
from sympy import isprime
def a(n):
k = 1
while not isprime(k*n+1): k += 1
return k
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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