Search: id:A092749
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%I A092749
%S A092749 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,
%T A092749 41,41,41,41,41,41,41,41,41,41,41,41,41,41,41
%N A092749 a(n) = least k such that m^2 + m + k is prime for m = 0, 1, ... n-1.
%C A092749 Comments from Pieter Moree (moree(AT)science.uva.nl), Apr 16 2004:
%C A092749 "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).
%C A092749 "Since the value of m^2+m+B is B^2 for m=B-1, one cannot possible do 
               better than this.
%C A092749 "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.
%C A092749 "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}."
%D A092749 Mollin, R. A. Prime-producing quadratics. Amer. Math. Monthly 104 (1997), 
               no. 6, 529-544.
%D A092749 Lukes, Patterson and Williams, (Numerical sieving devices: their history 
               and some applications, Nieuw Archief Wisk. 13 (1995), 113-139)
%e A092749 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.
%Y A092749 Cf. A014556.
%Y A092749 Sequence in context: A088887 A066911 A071850 this_sequence A152076 A138181 
               A133278
%Y A092749 Adjacent sequences: A092746 A092747 A092748 this_sequence A092750 A092751 
               A092752
%K A092749 nonn
%O A092749 1,1
%A A092749 Gabriel Cunningham (gcasey(AT)mit.edu), Apr 12 2004

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