cp's OEIS Frontend

A270752 (r,1)-greedy sequence, where r(k) = 1/(k*e).

%I A270752 #11 Mar 06 2025 22:13:54
%S A270752 1,1,1,1,1,1,1,1,262,167395,42355398928,2986137074379747535250,
%T A270752 16334453331070842795541380956715272941358931,
%U A270752 334377619479874433401339085661668551899899040409749812309411639875183486098285324762070
%N A270752 (r,1)-greedy sequence, where r(k) = 1/(k*e).
%C A270752 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 A270752 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 A270752 r(1)/a(1) + ... + r(n)/a(n) + ... = 1.
%e A270752 a(1) = ceiling(r(1)) = ceiling(1/e) = ceiling(0.367...) = 1;
%e A270752 a(2) = ceiling(r(2)/(1 - r(1)/1)) = 1;
%e A270752 a(3) = ceiling(r(3)/(1 - r(1)/1 - r(2)/1)) = 1.
%e A270752 The first 6 terms of the series r(1)/a(1) + ... + r(n)/a(n) + ... are 0.367..., 0.551..., 0.674..., 0.766..., 0.839..., 0.901...
%t A270752 $MaxExtraPrecision = Infinity; z = 16;
%t A270752 r[k_] := N[1/(k*E), 1000]; f[x_, 0] = x;
%t A270752 n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t A270752 f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t A270752 x = 1; Table[n[x, k], {k, 1, z}]
%t A270752 N[Sum[r[k]/n[x, k], {k, 1, 18}], 200]
%Y A270752 Cf. A001620, A270744, A068985.
%K A270752 nonn,easy
%O A270752 1,9
%A A270752 _Clark Kimberling_, Apr 09 2016