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A038198
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Numbers n such that n^2 + 7 is a power of 2.
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15
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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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
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LINKS
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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
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FORMULA
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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
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MATHEMATICA
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ok[n_] := Reduce[k>0 && n^2 + 7 == 2^k, k, Integers] =!= False; Select[Range[1000], ok] (* Jean-François Alcover, Sep 21 2011 *)
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CROSSREFS
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Cf. A060728, A002249, A107920, A227078.
Sequence in context: A154941 A062601 A231017 * A280876 A357055 A079037
Adjacent sequences: A038195 A038196 A038197 * A038199 A038200 A038201
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KEYWORD
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nonn,fini,full
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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