
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.
