

A076046


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


10




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


REFERENCES

Y. Bugeaud and T. Shorey. On the number of solutions of the generalized RamanujanNagell equation. J. reine angew. Math. 539 (2001), 5574.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", SpringerVerlag, 3rd, 1999. See Chapter 6.
H. Hasse. Uber eine diophantische Gleichung von RamanujanNagell und ihre Verallgemeinerung. Nagoya Math. J. 27 (1966), 77102.
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.


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.


CROSSREFS

Cf. A060728, A180445
Sequence in context: A016066 A012848 A202621 * A202380 A134807 A004002
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



