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Search: id:A092749
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| A092749 |
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a(n) = least k such that m^2 + m + k is prime for m = 0, 1, ... n-1. |
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4
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| 2, 3, 5, 5, 11, 11, 11, 11, 11, 11, 17, 17, 17, 17, 17, 17, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Comments from Pieter Moree (moree(AT)science.uva.nl), Apr 16 2004:
"The numbers 2,3,5,11,17 and 41 above are the only numbers B such that m^2+m+B is prime for m=0,....,B-2 (this can be proved, see Mollin's paper and is closely related to the celebrated Rabinowitsch criterion).
"Since the value of m^2+m+B is B^2 for m=B-1, one cannot possible do better than this.
"An obvious question of course is whether for given n, a(n) exists at all. This is far from obvious. Assuming the generally believed k-tuplets conjecture the answer is yes as was shown by Andrew Granville. For a proof (which is not very difficult) see the paper by Mollin.
"It is also known, due to work of Lukes, Patterson and Williams that any further elements in the above sequence, if they exist, are >10^{18}."
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REFERENCES
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Mollin, R. A. Prime-producing quadratics. Amer. Math. Monthly 104 (1997), no. 6, 529-544.
Lukes, Patterson and Williams, (Numerical sieving devices: their history and some applications, Nieuw Archief Wisk. 13 (1995), 113-139)
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EXAMPLE
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a(2) = 3 because 0^2 + 0 + 3 = 3 is prime and 1^2 + 1 + 3 = 5 is prime and it is the smallest number with the required properties.
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CROSSREFS
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Cf. A014556.
Sequence in context: A088887 A066911 A071850 this_sequence A152076 A138181 A133278
Adjacent sequences: A092746 A092747 A092748 this_sequence A092750 A092751 A092752
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KEYWORD
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nonn
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AUTHOR
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Gabriel Cunningham (gcasey(AT)mit.edu), Apr 12 2004
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