The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A038198 Numbers n such that n^2 + 7 is a power of 2. 15
 1, 3, 5, 11, 181 (list; graph; refs; listen; history; text; internal format)
 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 Ramanujan-Nagell 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) + ((1-i*sqrt(7))/2)^(k - 2))|. All terms furthermore follow form: |((1-i*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 1-i*sqrt(7))/2 figure prominently in the proof of the Ramanujan-Nagell Theorem (see below link, "The Ramanujan-Nagell 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 Spencer De Chenne, The Ramanujan-Nagell 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] (* Jean-François 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 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 21 03:10 EDT 2022. Contains 353886 sequences. (Running on oeis4.)