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 A076046 Ramanujan-Nagell numbers: the triangular numbers (of the form a*(a+1)/2) which are also of the form 2^b - 1. 12
 0, 1, 3, 15, 4095 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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 These 5 numbers are therefore the only ones which appear in column k=2 and also in the first subdiagonal of the Stirling2 Sheffer matrix S(n,k) = A048993(n,k). These entries are 0 = S(0, 2) = S(1, 2) = S(1, 0), 1 = S(2, 2) = S(2, 1), 3 = S(3, 2) (intersection of the column k=2 with the first subdiagonal), 15 = S(5, 2) = S(6, 5) and 4095 = S(13, 2) = S(91, 90). The motivation to look into this came from a comment of R. J. Cano on A247024. - Wolfdieter Lang, Oct 16 2014 REFERENCES J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd, 1999. See Chapter 6. T. Nagell. The Diophantine equation x^2 + 7 = 2^n. Nordisk Mat. Tidskr. 30 (1948), 62-64; Ark. Math. 4 (1960), 185-187. LINKS Y. Bugeaud and T. Shorey. On the number of solutions of the generalized Ramanujan-Nagell equation, J. reine angew. Math. 539 (2001), 55-74. H. Hasse, Uber eine diophantische Gleichung von Ramanujan-Nagell und ihre Verallgemeinerung, Nagoya Math. J. 27 (1966), 77-102. Eric Weisstein's World of Mathematics, Ramanujan's Square Equation. FORMULA 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 a(n) = ((sqrt(sqrt(A227079(n)) + 1)) * (sqrt(2) * (sqrt(sqrt(A227079(n)) + 1)) - 2))/(sqrt(32)). - Raphie Frank, Aug 08 2013 EXAMPLE 4095 can be written as 90*(90+1)/2, but also as 2^12 - 1. MATHEMATICA Reap[For[b = 0, b <= 12, b++, If[IntegerQ[(Sqrt[2^(b + 3) - 7] - 1)/2], Sow[2^b - 1]]]][[2, 1]] (* Jean-François Alcover, Jul 05 2017 *) Select[Accumulate[Range[0, 200]], IntegerQ[Log[2, #+1]]&] (* Harvey P. Dale, Aug 27 2019 *) CROSSREFS Cf. A060728, A038198, A180445, A215797. Sequence in context: A012848 A322714 A202621 * A202380 A290610 A134807 Adjacent sequences:  A076043 A076044 A076045 * A076047 A076048 A076049 KEYWORD fini,full,nonn AUTHOR Burt Totaro (b.totaro(AT)dpmms.cam.ac.uk), Oct 29 2002 STATUS approved

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Last modified October 23 09:45 EDT 2019. Contains 328345 sequences. (Running on oeis4.)