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%I #52 Jun 22 2021 12:29:33
%S 0,1,3,15,4095
%N Ramanujan-Nagell numbers: the triangular numbers (of the form a*(a+1)/2) which are also of the form 2^b - 1.
%C 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.
%C 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
%C Named after the Indian mathematician Srinivasa Ramanujan (1887-1920) and the Norwegian mathematician Trygve Nagell (1895-1988). - _Amiram Eldar_, Jun 22 2021
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd, 1999. See Chapter 6.
%D T. Nagell. The Diophantine equation x^2 + 7 = 2^n. Nordisk Mat. Tidskr., Vol. 30 (1948), pp. 62-64; Ark. Math., Vol. 4 (1960), pp. 185-187.
%H Yann Bugeaud and T. N. Shorey. <a href="http://dx.doi.org/10.1515/crll.2001.079">On the number of solutions of the generalized Ramanujan-Nagell equation</a>, J. reine angew. Math., Vol. 539 (2001), pp. 55-74.
%H Helmut Hasse, <a href="http://projecteuclid.org/euclid.nmj/1118801617">Uber eine diophantische Gleichung von Ramanujan-Nagell und ihre Verallgemeinerung</a>, Nagoya Math. J., Vol. 27 (1966), pp. 77-102.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujansSquareEquation.html">Ramanujan's Square Equation</a>.
%e 4095 can be written as 90*(90+1)/2, but also as 2^12 - 1.
%t 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 *)
%t Select[Accumulate[Range[0,200]],IntegerQ[Log[2,#+1]]&] (* _Harvey P. Dale_, Aug 27 2019 *)
%Y Cf. A060728, A038198, A180445, A215797.
%K fini,full,nonn
%O 1,3
%A Burt Totaro (b.totaro(AT)dpmms.cam.ac.uk), Oct 29 2002
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