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A076046
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Ramanujan-Nagell numbers: the triangular numbers (of the form a*(a+1)/2) which are also of the form 2^b - 1.
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8
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OFFSET
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1,3
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COMMENTS
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Ramanujan conjectured and Nagell proved, that the given numbers are the only ones. This sequence is equivalent to A060728, the list of numbers n such that x^2 + 7 = 2^n is soluble, by changing from n to 2^(n-3)-1.
2a(n)*{0,1,4,8,24} = Lambda_{0,1,4,8,24}, where Lambda_n is a laminated lattice Kissing number = {0,2,24,240,196560} = A215929. All elements of this sequence are "Sophie Germain triangular numbers of the first or second kind," as defined in A217278. Also see A216162, suggestive of a relationship between the two preceding comments and the Pell numbers. - Raphie Frank, Sep 30 2012
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REFERENCES
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Y. Bugeaud and T. Shorey. On the number of solutions of the generalized Ramanujan-Nagell equation. J. reine angew. Math. 539 (2001), 55-74.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd, 1999. See Chapter 6.
H. Hasse. Uber eine diophantische Gleichung von Ramanujan-Nagell und ihre Verallgemeinerung. Nagoya Math. J. 27 (1966), 77-102.
T. Nagell. The Diophantine equation x^2 + 7 = 2^n. Nordisk Mat. Tidskr. 30 (1948), 62-64; Ark. Math. 4 (1960), 185-187.
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LINKS
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Table of n, a(n) for n=1..5.
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FORMULA
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a(n) = (2^(A060728(n) - 3) - 1) = 1/2*((ceiling[sqrt(2^(A060728(n) - 2) - 1)])^2 - (ceiling[sqrt(2^(A060728(n) - 2) - 1)])) = 1/2*(A180445(n)^2 - A180445(n)) - Raphie Frank, Mar 31 2013
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EXAMPLE
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4095 can be written as 90*(90+1)/2, but also as 2^12 - 1.
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CROSSREFS
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Cf. A060728, A180445
Sequence in context: A016066 A012848 A202621 * A202380 A134807 A004002
Adjacent sequences: A076043 A076044 A076045 * A076047 A076048 A076049
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KEYWORD
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fini,full,nonn
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AUTHOR
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Burt Totaro (b.totaro(AT)dpmms.cam.ac.uk), Oct 29 2002
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STATUS
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approved
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