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A270748 (r,1)-greedy sequence, where r(k) = 2/log(k+1).

%I A270748 #13 Mar 06 2025 08:18:22
%S A270748 3,48,5215,43930979,8221176288381971,
%T A270748 237472642129791861355082716048930,
%U A270748 59916111345562665920456160598356741759066440491193682529746704653
%N A270748 (r,1)-greedy sequence, where r(k) = 2/log(k+1).
%C A270748 Let x > 0, and let r = (r(k)) be a sequence of positive irrational numbers.  Let a(1) be the least positive integer m such that r(1)/m < x, and inductively let a(n) be the least positive integer m such that r(1)/a(1) + ... + r(n-1)/a(n-1) + r(n)/m < x.  The sequence (a(n)) is the (r,x)-greedy sequence.  We are interested in choices of r and x for which the series r(1)/a(1) + ... + r(n)/a(n) + ... converges to x.  See A270744 for a guide to related sequences.
%F A270748 a(n) = ceiling(r(n)/s(n)), where s(n) = 1 - r(1)/a(1) - r(2)/a(2) - ... - r(n-1)/a(n-1).
%F A270748 r(1)/a(1) + ... + r(n)/a(n) + ... = 1.
%e A270748 a(1) = ceiling(r(1)) = ceiling(2/log(2)) = ceiling(2.885...) = 3;
%e A270748 a(2) = ceiling(r(2)/(1 - r(1)/3)) = 48;
%e A270748 a(3) = ceiling(r(3)/(1 - r(1)/3 - r(2)/48)) = 5215.
%e A270748 The first 3 terms of the series r(1)/a(1) + ... + r(n)/a(n) + ... are 0.961..., 0.997..., 0.99999997...
%t A270748 $MaxExtraPrecision = Infinity; z = 16;
%t A270748 r[k_] := N[2/Log[k + 1], 1000]; f[x_, 0] = x;
%t A270748 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270748 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270748 x = 1; Table[n[x, k], {k, 1, z}]
%t A270748 N[Sum[r[k]/n[x, k], {k, 1, 18}], 200]
%Y A270748 Cf. A001620, A270744.
%K A270748 nonn,easy
%O A270748 1,1
%A A270748 _Clark Kimberling_, Apr 09 2016