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 A278448 a(n) = nearest integer to b(n) = c^(b(n-1)/(n-1)), where c=2 and b(1) is chosen such that the sequence neither explodes nor goes to 1. 7
 3, 7, 13, 19, 25, 32, 39, 46, 53, 61, 69, 77, 85, 93, 102, 110, 119, 128, 136, 145, 154, 163, 173, 182, 191, 201, 210, 220, 229, 239, 248, 258, 268, 278, 288, 298, 307, 318, 328, 338, 348, 358, 368, 379, 389, 399, 410, 420, 430, 441, 451, 462, 473, 483, 494, 505, 515, 526, 537, 547, 558, 569, 580, 591, 602, 613, 624, 635, 646, 657 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS For the given c there exists a unique b(1) for which the sequence b(n) does not converge to 1 and at the same time always satisfies b(n-1)b(n+1)/b(n)^2 < 1 (due to rounding to the nearest integer a(n-1)a(n+1)/a(n)^2 is not always less than 1). In this case b(1) = 2.8718808270... A278808. If b(1) were chosen smaller the sequence would approach 1, if it were chosen greater the sequence would at some point violate b(n-1)b(n+1)/b(n)^2 < 1 and from there on quickly escalate. The value of b(1) is found through trial and error. Suppose one starts with b(1) = 2, the sequence would continue b(2) = 4, b(3) = 4, b(4) = 2.51..., b(5) = 1.54... and from there one can see that such a sequence is tending to 1. One continues by trying a larger value, say b(1) = 3, which gives rise to b(2) = 8, b(3) = 16, b(4) = 40.31... and from there one can see that such a sequence is escalating too fast. Therefore, one now knows that the true value of b(1) is between 2 and 3. b(n) = n*log_2((n+1)*log_2((n+2)*log_2(...))) ~ n*log_2(n). - Andrey Zabolotskiy, Dec 01 2016 LINKS Rok Cestnik, Table of n, a(n) for n = 1..1000 Rok Cestnik, Plot of the dependence of b(1) on c EXAMPLE a(2) = round(2^2.87...) = round(7.32...) = 7. a(3) = round(2^(7.32.../2)) = round(12.64...) = 13. a(4) = round(2^(12.64.../3)) = round(18.55...) = 19. MATHEMATICA c = 2; n = 100; acc = Round[n*1.2]; th = 1000000; b1 = 0; For[p = 0, p < acc, ++p,   For[d = 0, d < 9, ++d,     b1 = b1 + 1/10^p;     bn = b1;     For[i = 1, i < Round[n*1.2], ++i,      bn = N[c^(bn/i), acc];      If[bn > th, Break[]];      ];     If[bn > th, {       b1 = b1 - 1/10^p;       Break[];       }];     ];   ]; bnlist = {N[b1]}; bn = b1; For[i = 1, i < n, ++i,   bn = N[c^(bn/i), acc];   If[bn > th, Break[]];   bnlist = Append[bnlist, N[bn]];   ]; anlist = Map[Round[#] &, bnlist] CROSSREFS For decimal expansion of b(1) see A278808. For different values of c see A278449, A278450, A278451, A278452. For b(1)=0 see A278453. Sequence in context: A144917 A102828 A296510 * A117679 A310265 A276215 Adjacent sequences:  A278445 A278446 A278447 * A278449 A278450 A278451 KEYWORD nonn AUTHOR Rok Cestnik, Nov 22 2016 STATUS approved

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Last modified January 25 19:09 EST 2020. Contains 331249 sequences. (Running on oeis4.)