A336881 a(n) is the number of solutions (x, m) of the generalized Ramanujan-Nagell equation x^2 + n = 2^m, x > 0, m > 0, n > 0.

1, 0, 1, 1, 0, 0, 5, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 5, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Bernard Schott, Aug 06 2020

Keywords

• nonn

Comments

Equivalently, number of representations of n as n = 2^m - x^2, m > 0, x > 0.

a(7) = 5 corresponds to Ramanujan-Nagell equation (A038198 for x, A060728 for m, Wikipedia link).

If n odd <> 7, Apéry proved in 1960 that the equation x^2 + n = 2^m has at most 2 solutions (see link).

If n odd, this equation has 2 solutions iff n = 23 or n = 2^k - 1 for some k >= 4 (link Beukers, theorem 2, p. 395).

Examples

			1^2 + 1 = 2^1 hence a(1) = 1.
3^2 + 23 = 2^5 and 45^2 + 23 = 2^11 hence a(23) = 2.
28 = 2^5 - 2^2 = 2^6 - 6^2 = 2^7 - 10^2 = 2^9 - 22^2 = 2^17 - 362^2 hence a(28) = 5.
		

Links

• Roger Apéry, Sur une équation Diophantienne, C. R. Acad. Sci. Paris Sér. A251 (1960), 1263-1264.
• Frits Beukers, On the generalized Ramanujan-Nagell equation, I, Acta arithmetica, XXXVIII, 1980-1981, page 389-410.
• Wikipedia, Ramanujan-Nagell equation.

Crossrefs

Cf. A247763, A336819.

Extensions

More terms from Jinyuan Wang, Aug 07 2020