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 A086378 Sum of successive remainders in computing Euclidean algorithm for (1,1/sqrt(n)) is rational. 5
 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 55, 56, 64, 70, 72, 81, 90, 100, 110, 121, 132, 144, 153, 155, 156, 169, 180, 182, 196, 210, 225, 240, 256, 272, 289, 305, 306, 324, 342, 361, 377, 380, 400, 420, 441, 462, 484, 504, 505, 506, 529, 546, 552, 576, 600 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS For a real number x, take (a_0,b_0) = (1,x), then (a_(i+1),b_(i+1)) = (b_i,a_i-b_i*floor(a_i/b_i)), for i>=0 and call kappa(x) = b_1+b_2+b_3+... If kappa(1/sqrt(n)) is rational (which can be easily evaluated thanks to the periodicity of the process for a quadratic number), then n is in the sequence. An infinity of 2nd degree polynomial functions take all their values over N in the sequence (such as x^2, x^2+x, 36*x^2+17*x+2, 100*x^2+150*x+55, 196*x^2+97*x+12, ...). LINKS EXAMPLE 7137 is in the sequence because kappa(1/sqrt(7137)) = 7/1098 (in Q). MATHEMATICA kappa[n_] := Module[{a, b, i, p}, If[(a = Sqrt[n] - Floor[Sqrt[n]]) == 0, Return]; i = a = Simplify[1/a]; p = 1; b = 0; Do[p = Simplify[a*p]; b = Simplify[a*b - Floor[a] + a]; If[(a = Simplify[1/(a - Floor[a])]) == i, Break[]], {Infinity}]; Return[Simplify[(1/a + b/(p-1))/Sqrt[n], Sqrt]]]; Reap[For[n = 1, n <= 600, n++, If[Element[kappa[n], Rationals], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Apr 15 2017, translated from MuPAD *) PROG (MuPAD) kappa_1_over_sqrt := proc(n) local a, b, i, p; begin if (a := sqrt(n)-isqrt(n)) = 0 then return(0) end_if: i := a := simplify(1/a, sqrt); p := 1; b := 0; repeat p := simplify(p*a, sqrt); b := simplify(b*a+a-floor(a), sqrt); until (a := simplify(1/(a-floor(a)), sqrt)) = i end_repeat: return(simplify((b/(p-1) + 1/a)/sqrt(n), sqrt)); end_proc: CROSSREFS Sequence in context: A075354 A194164 A194254 * A088900 A083392 A076921 Adjacent sequences:  A086375 A086376 A086377 * A086379 A086380 A086381 KEYWORD nonn AUTHOR Thomas Baruchel, Sep 02 2003 EXTENSIONS Edited by Franklin T. Adams-Watters, Nov 30 2011 STATUS approved

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Last modified October 20 12:37 EDT 2021. Contains 348107 sequences. (Running on oeis4.)