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A034693
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Smallest k such that k*n+1 is prime.
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32
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Conjecture: for every n there exists a number k < n such that n*k + 1 is a prime - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), 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 (joseph_l_pe(AT)hotmail.com), Jul 16 2002
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REFERENCES
| S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 127-130.
Graham, D. (1981): On Linnik's Constant. Acta Arithm., 39, pp. 163-179.
Niven I. and Powell B (1976): Primes in Certain Arithmetic Progressions. Amer. Math. Monthly,83:467-489.
Ribenboim, P. (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
| T. D. Noe, Table of n, a(n) for n=1..10000
S. R. Finch, More about Linnik's Constant
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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
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EXAMPLE
| If n=7, the smallest prime in the sequence 8,15,22,29,... is 29, so a(7)=4.
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MATHEMATICA
| a[n_]:=(k=0; While[!PrimeQ[++k*n + 1]]; k); Table[a[n], {n, 100}] (* From Jean-François Alcover, Jul 19 2011 *)
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PROG
| (PARI) a(n)=if(n<0, 0, s=1; while(isprime(s*n+1)==0, s++); s)
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CROSSREFS
| Cf. A034694.
Cf. A085420.
Sequence in context: A100380 A205403 A080825 * A072342 A066389 A077191
Adjacent sequences: A034690 A034691 A034692 * A034694 A034695 A034696
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KEYWORD
| nonn,nice
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AUTHOR
| Labos Elemer (LABOS(AT)ana.sote.hu)
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