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A080059 Greedy powers of (1/zeta(3)): Sum_{n>=1} (1/zeta(3))^a(n) = 1, where 1/zeta(3) = .83190737258070746868...

%I A080059 #10 Sep 01 2025 22:55:43
%S A080059 1,10,26,38,54,64,80,98,115,126,136,147,158,171,181,196,206,226,243,
%T A080059 257,267,279,293,306,324,334,355,365,378,388,398,410,432,442,455,468,
%U A080059 491,501,519,534,545,560,572,582,593,610,628,638,650,663,672,691,704,715
%N A080059 Greedy powers of (1/zeta(3)): Sum_{n>=1} (1/zeta(3))^a(n) = 1, where 1/zeta(3) = .83190737258070746868...
%C A080059 The n-th greedy power of x, when 0.5 < x < 1, is the smallest integer exponent a(n) that does not cause the power series Sum_{k=1..n} x^a(k) to exceed unity. A heuristic argument suggests that the limit of a(n)/n is m - Sum_{n>=m} log(1 + x^n)/log(x) = 14.874449248373..., where x=(1/zeta(3)) and m=floor(log(1-x)/log(x))=9.
%C A080059 See A077468 for Mathematica program by _Robert G. Wilson v_.
%F A080059 a(n) = Sum_{k=1..n} floor(g_k) where g_1=1, g_{n+1}=log_x(x^frac(g_n) - x) (n>0) at x=(1/zeta(3)) and frac(y) = y - floor(y).
%e A080059 a(3)=26 since (1/zeta(3)) +(1/zeta(3))^10 +(1/zeta(3))^26 < 1 and (1/zeta(3)) +(1/zeta(3))^10 +(1/zeta(3))^k > 1 for 10<k<26.
%Y A080059 Cf. A077468, A080059.
%K A080059 nonn,changed
%O A080059 1,2
%A A080059 _Benoit Cloitre_ and _Paul D. Hanna_, Jan 23 2003