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 A248559 Least k such that log(2) - sum{1/(h*2^h), h = 1..k} < 1/3^n. 5
 1, 2, 3, 4, 6, 7, 8, 10, 11, 13, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 74, 75, 77, 78, 80, 81, 83, 84, 86, 88, 89, 91, 92, 94, 95, 97, 98 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This sequence provides insight into the manner of convergence of  sum{1/(h*2^h), h = 1..k} to log 2.  Since a(n+1) - a(n) is in {1,2} for n >= 1, the sequences A248560 and A248561 partition the positive integers. REFERENCES Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 15. LINKS Clark Kimberling, Table of n, a(n) for n = 1..1000 EXAMPLE Let s(n) = log(2) - sum{1/(h*2^h), h = 1..n}.  Approximations follow: n ... s(n) ........ 1/3^n 1 ... 0.193147 .... 0.33333 2 ... 0.0681472 ... 0.11111 3 ... 0.0264805 ... 0.037037 4 ... 0.0108555 ... 0.0123457 5 ... 0.0046066 ... 0.004115 6 ... 0.0020013 ... 0.00137174 a(5) = 6 because s(6) < 1/3^5 < s(5). MATHEMATICA z = 200; p[k_] := p[k] = Sum[1/(h*2^h), {h, 1, k}] N[Table[Log[2] - p[n], {n, 1, z/5}]] f[n_] := f[n] = Select[Range[z], Log[2] - p[#] < 1/3^n &, 1] u = Flatten[Table[f[n], {n, 1, z}]]    (* A248559 *) Flatten[Position[Differences[u], 1]]   (* A248560 *) Flatten[Position[Differences[u], 2]]   (* A248561 *) CROSSREFS Cf. A002162 (log(2)), A248560, A248561. Sequence in context: A284941 A354767 A351714 * A186511 A066049 A160697 Adjacent sequences:  A248556 A248557 A248558 * A248560 A248561 A248562 KEYWORD nonn,easy AUTHOR Clark Kimberling, Oct 09 2014 STATUS approved

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Last modified August 8 21:55 EDT 2022. Contains 356016 sequences. (Running on oeis4.)