

A076046


RamanujanNagell numbers: the triangular numbers (of the form a*(a+1)/2) which are also of the form 2^b  1.


12




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^(n3)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", SpringerVerlag, 3rd, 1999. See Chapter 6.
T. Nagell. The Diophantine equation x^2 + 7 = 2^n. Nordisk Mat. Tidskr. 30 (1948), 6264; Ark. Math. 4 (1960), 185187.


LINKS

Table of n, a(n) for n=1..5.
Y. Bugeaud and T. Shorey. On the number of solutions of the generalized RamanujanNagell equation, J. reine angew. Math. 539 (2001), 5574.
H. Hasse, Uber eine diophantische Gleichung von RamanujanNagell und ihre Verallgemeinerung, Nagoya Math. J. 27 (1966), 77102.
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]] (* JeanFrançois 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



