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 A060728 Numbers n such that Ramanujan's equation x^2 + 7 = 2^n has an integer solution. 15
 3, 4, 5, 7, 15 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS See A038198 for corresponding x. - Lekraj Beedassy, Sep 07 2004 Also numbers such that 2^(n-3)-1 is in A000217, i.e., a triangular number. - M. F. Hasler, Feb 23 2009 With respect to M. F. Hasler's comment above, all terms 2^(n-3) - 1 are known as the Ramanujan-Nagell triangular numbers (A076046). - Raphie Frank, Mar 31 2013 REFERENCES J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 181, p. 56, Ellipses, Paris 2008. J. Roberts, Lure of the Integers. pp. 90-91, MAA 1992. Ian Stewart & David Tall, Algebraic Number Theory and Fermat's Last Theorem, 3rd Ed. Natick, Massachusetts (2002): 96-98. LINKS T. Skolem, S. Chowla and D. J. Lewis, The Diophantine Equation 2^(n+2)-7=x^2 and Related Problems. Proc. Amer. Math. Soc. 10 (1959) 663-669. [M. F. Hasler, Feb 23 2009] M. Beeler, R. W. Gosper and R. Schroeppel, HAKMEM: item 31: A Ramanujan Problem (R. Schroeppel) Curtis Bright, Solving Ramanujan's Square Equation Computationally Spencer De Chenne, The Ramanujan-Nagell Theorem: Understanding the Proof A. Engel, Problem-Solving Strategies. p. 126. Gerry Myerson, Bibliography T. Nagell, The Diophantine equation x^2 + 7 = 2^n, Ark. Mat. 4 (1961), no. 2-3, 185-187. S. Ramanujan, Journal of the Indian Mathematical Society, Question 464(v,120) Eric Weisstein's World of Mathematics, Ramanujan's Square Equation Eric Weisstein's World of Mathematics, Diophantine Equation 2nd Powers Wikipedia, Carmichael's Theorem Wikipedia, Diophantine equation FORMULA a(n) = log_2(8*A076046(n) + 8) = log_2(A227078(n) + 7) Empirically, a(n) = Fibonacci(c + 1) + 2 = ceiling[e^((c - 1)/2)] + 2 where {c} is the complete set of positive solutions to {n in N | 2 cos(2*Pi/n) is in Z}; c is in {1,2,3,4,6} (see A217290). EXAMPLE The fifth and ultimate solution to Ramanujan's equation is obtained for the 15th power of 2, so that we have x^2 + 7 = 2^15 yielding x = 181. MATHEMATICA ramaNagell[n_] := Reduce[x^2 + 7 == 2^n, x, Integers] =!= False; Select[ Range[100], ramaNagell] (* Jean-François Alcover, Sep 21 2011 *) PROG (MAGMA) [n: n in [0..100] | IsSquare(2^n-7)]; // Vincenzo Librandi, Jan 07 2014 (PARI) is(n)=issquare(2^n-7) \\ Anders Hellström, Dec 12 2015 CROSSREFS Cf. A002249, A038198, A076046, A077020, A077021, A107920, A215795, A227078 Sequence in context: A089560 A248077 A239547 * A295988 A216433 A101761 Adjacent sequences:  A060725 A060726 A060727 * A060729 A060730 A060731 KEYWORD fini,full,nonn AUTHOR Lekraj Beedassy, Apr 25 2001 EXTENSIONS Added keyword "full", M. F. Hasler, Feb 23 2009 STATUS approved

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Last modified October 22 00:52 EDT 2019. Contains 328315 sequences. (Running on oeis4.)