A038198 Numbers n such that n^2 + 7 is a power of 2.

1, 3, 5, 11, 181

Offset

1,2

Comments

The exponents of the corresponding powers of 2 are 3, 4, 5, 7, 15 (see Ramanujan). - N. J. A. Sloane, Jun 01 2014

The terms lead to identities resembling Machin's Pi/4 = arctan(1/1) = 4*arctan(1/5) - arctan(1/239), for example, arctan(sqrt(7)/1) = 5*arctan(sqrt(7)/11) + 2*arctan(sqrt(7)/181), which can also be expressed as arcsin(sqrt(7/2^3)) = 5*arcsin(sqrt(7/2^7)) + 2*arcsin(sqrt(7/2^15)) (cf. A168229). - Joerg Arndt, Nov 09 2012

References

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 181, p. 56, Ellipses, Paris 2008.

L. J. Mordell, Diophantine Equations, Academic Press, NY, 1969, p. 205.

S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962. See Question 464, p. 327. - N. J. A. Sloane, Jun 01 2014

Links

Table of n, a(n) for n=1..5.

Spencer De Chenne, The Ramanujan-Nagell Theorem: Understanding the Proof

Eric Weisstein's World of Mathematics, Ramanujan's Square Equation

Wikipedia, Lucas Sequence

Mathematica

ok[n_] := Reduce[k>0 && n^2 + 7 == 2^k, k, Integers] =!= False; Select[Range[1000], ok] (* Jean-François Alcover, Sep 21 2011 *)

Prog

(PARI) [x | n<-[0..99], issquare(2^n-7, &x)] \\ M. F. Hasler, Mar 11 2024

Crossrefs

Cf. A060728, A002249, A107920.

Sequence in context: A154941 A062601 A231017 * A280876 A357055 A079037

Adjacent sequences: A038195 A038196 A038197 * A038199 A038200 A038201

Keyword

nonn,fini,full

Author

N. J. A. Sloane

Status

approved