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Search: id:A034693
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| A034693 |
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Smallest k such that kn+1 is prime. |
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+0
28
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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; listen)
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OFFSET
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1,3
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COMMENT
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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
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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
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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
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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 (benoit7848c(AT)orange.fr), 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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PROGRAM
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(PARI) a(n)=if(n<0, 0, s=1; while(isprime(s*n+1)==0, s++); s)
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CROSSREFS
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Cf. A034694.
Cf. A085420.
Sequence in context: A068341 A100380 A080825 this_sequence A072342 A066389 A077191
Adjacent sequences: A034690 A034691 A034692 this_sequence A034694 A034695 A034696
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KEYWORD
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nonn,nice
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AUTHOR
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Labos Elemer (LABOS(AT)ana.sote.hu)
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