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A270750 (r,1)-greedy sequence, where r(k) = 1/(k*log(k+1)).

%I A270750 #13 Mar 06 2025 08:18:04
%S A270750 2,2,5,52,7132,657650603,642344866115572775,
%T A270750 833790618410287382945149122154404558,
%U A270750 1229679779588111283437146138551802288646488858072438842199407751052675116
%N A270750 (r,1)-greedy sequence, where r(k) = 1/(k*log(k+1)).
%C A270750 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 A270750 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 A270750 r(1)/a(1) + ... + r(n)/a(n) + ... = 1.
%e A270750 a(1) = ceiling(r(1)) = ceiling(1/log(2)) = ceiling(1.442...) = 2;
%e A270750 a(2) = ceiling(r(2)/(1 - r(1)/2)) = 2;
%e A270750 a(3) = ceiling(r(3)/(1 - r(1)/2 - r(2)/2)) = 5.
%e A270750 The first 3 terms of the series r(1)/a(1) + ... + r(n)/a(n) + ... are 0.721..., 0.948..., 0.996...
%t A270750 $MaxExtraPrecision = Infinity; z = 16;
%t A270750 r[k_] := N[1/(k*Log[k + 1]), 1000]; f[x_, 0] = x;
%t A270750 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270750 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270750 x = 1; Table[n[x, k], {k, 1, z}]
%t A270750 N[Sum[r[k]/n[x, k], {k, 1, 18}], 200]
%Y A270750 Cf. A001620, A270744.
%K A270750 nonn,easy
%O A270750 1,1
%A A270750 _Clark Kimberling_, Apr 09 2016