

A038198


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


13




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
These terms squared are the RamanujanNagell squares (A227078(n) = 2^(A060728(n)  7)). Where k is in A060728 = {3, 4, 5, 7, 15}, all terms also follow form: (((1+i*sqrt(7))/2)^(k  2) + ((1i*sqrt(7))/2)^(k  2)). All terms furthermore follow form: ((1i*sqrt(7))^(2*k  4)(1+i*sqrt(7))^(2*k  4))*i/(2^(2*k  4)*sqrt(7)). This follows from the properties of Lucas sequences as demonstrated in the formula section below. These formulas are interesting since the forms 1+i*sqrt(7))/2 and 1i*sqrt(7))/2 figure prominently in the proof of the RamanujanNagell Theorem (see below link, "The RamanujanNagell Theorem: Understanding the Proof").  Raphie Frank, Dec 25 2013


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 RamanujanNagell Theorem: Understanding the Proof
Eric Weisstein's World of Mathematics, Ramanujan's Square Equation
Wikipedia, Lucas Sequence


FORMULA

a(n) = sqrt(2^A060728(n)  7). Alternatively, where A002249(n) = V_n(P, Q) = V_n(1, 2) and A107920(n) = U_n(P, Q) = U_n(1, 2), then a(n) = A002249(A060728(n)  2) = A002249(A060728(n)  2)* A107920(A060728(n)  2) = A107920(2*A060728(n)  4). Note that A107920(A060728(n)  2) is 1, which is why this equivalency holds (U_2n = U_n*V_n).  Raphie Frank, Dec 25 2013


MATHEMATICA

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


CROSSREFS

Cf. A060728, A002249, A107920, A227078.
Sequence in context: A154941 A062601 A231017 * A280876 A079037 A281087
Adjacent sequences: A038195 A038196 A038197 * A038199 A038200 A038201


KEYWORD

nonn,fini,full


AUTHOR

N. J. A. Sloane


STATUS

approved



