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Search: id:A005846
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| A005846 |
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Primes of form n^2 + n + 41.
(Formerly M5273)
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+0
35
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| 41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601, 1847, 1933, 2111, 2203, 2297, 2393, 2591, 2693, 2797
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Note that 41 is the largest of Euler's Lucky numbers (A014556). - Lekraj Beedassy (blekraj(AT)yahoo.com), Apr 22 2004
a(n)=A117530(13,n) for n<=13: a(1)=A117530(13,1)=A014556(6)=41, A117531(13)=13. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 26 2006
The g.f. -(41-80*z+41*z**2)/(z-1)**3 conjectured by S. Plouffe in his 1992 dissertation is wrong.
The link to E. Wegrzynowski contents the following false statement: "It is possible to find a polynomial of the form n^2 + n + B that gives prime numbers for n = 0...A, A being any number." It is known that the maximum is A = 39 for B = 41. - Luis Rodriguez (luiroto(AT)yahoo.com), Jun 22 2008
Contrary to the last comment, Mollin's Theorem 2.1 shows that any A is possible if the Prime k-tuples Conjecture is assumed. [From T. D. Noe (noe(AT)sspectra.com), Aug 31 2009]
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 137.
R. A. Mollin, Prime producing quadratics, Amer. Math. Monthly 104 (1997), 529-544. [From T. D. Noe (noe(AT)sspectra.com), Aug 31 2009]
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LINKS
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Zak Seidov, Table of n, a(n) for n = 1..10000.
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
E. Wegrzynowski, Les formules simples qui donnent des nombres premiers en grande quantite
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
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a(n) =A056561(n)^2+A056561(n)+41
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EXAMPLE
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a(39)=1601=39^2+39+41 is in the sequence because it is prime. 1681=40^2+40+41 is not because 1681=41*41.
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MATHEMATICA
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lst={}; Do[p=n^2+n+41; If[PrimeQ[p], AppendTo[lst, p]], {n, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 05 2009]
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CROSSREFS
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Cf. A048988, A007634, A056561, A002378, A007635.
Sequence in context: A118124 A054057 A155884 this_sequence A154498 A062669 A045710
Adjacent sequences: A005843 A005844 A005845 this_sequence A005847 A005848 A005849
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Henry Bottomley (se16(AT)btinternet.com), Jun 26 2000
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