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A248562
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Least k such that log(3/2) - sum{1/(h*3^h), h = 1..k} < 1/6^n.
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5
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1, 2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 28, 29, 31, 33, 34, 36, 37, 39, 41, 42, 44, 45, 47, 49, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 73, 74, 76, 77, 79, 81, 82, 84, 86, 87, 89, 90, 92, 94, 95, 97, 98, 100, 102, 103
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OFFSET
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1,2
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COMMENTS
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This sequence provides insight into the manner of convergence of sum{1/(h*3^h), h = 1..k} to log(3/2). Since a(n+1) - a(n) is in {1,2} for n >= 1, the sequences A248563 and A248564 partition the positive integers.
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 15.
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LINKS
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EXAMPLE
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Let s(n) = log(3/2) - sum{1/(h*3^h), h = 1..n}. Approximations follow:
n ... s(n) ........ 1/6^n
1 ... 0.0721318 ... 0.166667
2 ... 0.0165762 ... 0.0277777
3 ... 0.0042305 ... 0.0046296
4 ... 0.0011441 ... 0.0007716
5 ... 0.0003210 ... 0.0001286
a(4) = 5 because s(5) < 1/6^4 < s(4).
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MATHEMATICA
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z = 300; p[k_] := p[k] = Sum[1/(h*3^h), {h, 1, k}];
N[Table[Log[3/2] - p[n], {n, 1, z/5}]]
f[n_] := f[n] = Select[Range[z], Log[3/2] - p[#] < 1/6^n &, 1];
u = Flatten[Table[f[n], {n, 1, z}]] (* A248562 *)
Flatten[Position[Differences[u], 1]] (* A248563 *)
Flatten[Position[Differences[u], 2]] (* A248564 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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