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A177435
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The smallest prime p such that p//9_(n), its concatenation with n nines, is prime.
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1
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2, 5, 2, 5, 5, 2, 2, 7, 41, 5, 41, 41, 5, 23, 101, 13, 17, 29, 2, 37, 13, 5, 5, 79, 7, 19, 2, 5, 131, 47, 47, 103, 59, 5, 43, 89, 149, 71, 17, 5, 193, 29, 2, 277, 73, 107, 127, 79, 7, 103, 83, 19, 173, 29, 2, 23, 107, 103, 43, 47, 5, 13, 73, 503, 307, 197, 83, 113, 331, 163
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OFFSET
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1,1
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COMMENTS
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a(n) is the smallest prime p such that 10^n*(p+1)-1 is a prime number.
(a) if a(n) = 29 then a(n+1) = 2, (b) if a(n) = 59 and a(n+1) > 2 then a(n+1) = 5, etc.
(c) if a(n) = 19 then a(n+1) >= 2, as 1 is not a prime.
The smallest prime that is not appearing as such a concatenation seems to be 11:
The cases 11//9_(n) with n even, that is 11//9_(2*k), are multiples of 11 and ruled out.
Most cases 11//9_(n) with n odd are also ruled out:
11//9_(6*k+1) are multiples of 7. 11//9_(6*k+3) are multiples of 13.
It seems where 11//9_(6*k+5) is prime, so is 5//9_(6*k+5).
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REFERENCES
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M. du Sautoy: Die Musik der Primzahlen: Auf den Spuren des groessten Raetsels der Mathematik, Deutscher Taschenbuch Verlag, 2006
F. Ischebeck: Einladung zur Zahlentheorie, B.I. Wissenschaftsverlag, Mannheim-Leipzig-Wien-Zuerich, 1992
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LINKS
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EXAMPLE
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2//9 = 29 = prime(10), 2 = prime(1) is a(1).
5//99 = 599 = prime(109), 5 = prime(3) is a(2).
163//9_(70) is prime, so a(70)=163.
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 08 2010
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EXTENSIONS
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Used variable n for the index; used more cautious wording for the unproved n=6k+5 case for the 11's - R. J. Mathar, May 10 2010
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STATUS
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approved
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