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A224868
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a(1) = greatest k such that H(k) - H(4) < 1/3 + 1/4; a(2) = greatest k such that H(k) - H(a(1)) < H(a(1)) - H(4); and for n > 2, a(n) = greatest k such that H(k) - H(a(n-1)) > H(a(n-1)) - H(a(n-2)), where H = harmonic number.
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5
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7, 11, 17, 26, 39, 58, 86, 127, 187, 275, 404, 593, 870, 1276, 1871, 2743, 4021, 5894, 8639, 12662, 18558, 27199, 39863, 58423, 85624, 125489, 183914, 269540, 395031, 578947, 848489, 1243522, 1822471, 2670962, 3914486, 5736959, 8407923, 12322411, 18059372
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OFFSET
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1,1
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COMMENTS
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Suppose that x and y are positive integers and that x <=y. Let c(1) = y and c(2) = greatest k such that H(k) - H(y) < H(y) - H(x); for n > 2, let c(n) = greatest such that H(k) - H(c(n-1)) < H(c(n-1)) - H(c(n-2)). Then 1/x + ... + 1/c(1) > 1/(c(1)+1) + ... + 1/(c(2)) > 1/(c(2)+1) + ... + 1/(c(3)) > ... The decreasing sequences H(c(n)) - H(c(n-1)) and c(n)/c(n-1) converge. For what choices of (x,y) is the sequence c(n) linearly recurrent?
For A224868, (x,y) = (3,4); it appears that the sequence a(n) is linearly recurrent with signature (2,-1,1,-1). Possibly the constant at A202537 is the limit of the sequences H(c(n))-H(c(n-1)). Possibly the constant at A092526 is the limit of c(n)/c(n-1).
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LINKS
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FORMULA
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a(n) = 2*a(n-1) - a(n-2) + a(n-3) - a(n-4) (conjectured).
G.f.: (7 - 3 x + 2 x^2 - 4 x^3)/(1 - 2 x + x^2 - x^3 + x^4) (conjectured).
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EXAMPLE
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The first three values (a(1),a(2),a(3)) = (7,11,17) match the beginning of the following inequality chain (and partition of {1/m: m>=3}):
1/3+1/4 > 1/5+1/6+1/7 > 1/8+1/9+1/10+1/11 > 1/12+ ... +1/17 > ...
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MATHEMATICA
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z = 100; h[n_] := h[n] = HarmonicNumber[N[n, 500]]; x = 3; y = 4; a[1] = -1 + Ceiling[w /. FindRoot[h[w] == 2 h[y] - h[x - 1], {w, 1}, WorkingPrecision -> 400]]; a[2] = -1 + Ceiling[w /. FindRoot[h[w] == 2 h[a[1]] - h[y], {w, a[1]}, WorkingPrecision -> 400]]; Do[s = 0; a[t] = -1 + Ceiling[w /. FindRoot[h[w] == 2 h[a[t - 1]] - h[a[t - 2]], {w, a[t - 1]}, WorkingPrecision -> 400]], {t, 3, z}]; m = Map[a, Range[z]] (* A224868 *)
N[Table[h[a[t]] - h[a[t - 1]], {t, 2, z, 25}], 5] (* A202537? *)
N[Table[a[n]/a[n - 1], {n, 2, z, 25}], 5] (* A092526? *)
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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STATUS
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approved
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