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A080058 Greedy powers of (1/zeta(2)): Sum_{n>=1} (1/zeta(2))^a(n) = 1, where 1/zeta(2) = 6/Pi^2 = .607927101854...

%I A080058 #14 Sep 01 2025 22:52:39
%S A080058 1,2,8,12,14,16,25,39,42,44,46,49,51,53,59,70,73,78,81,83,85,86,101,
%T A080058 103,105,116,118,119,126,130,135,137,139,142,144,147,148,158,161,163,
%U A080058 170,171,178,181,186,188,190,192,194,195,204,207,209,212,216,219,224,229
%N A080058 Greedy powers of (1/zeta(2)): Sum_{n>=1} (1/zeta(2))^a(n) = 1, where 1/zeta(2) = 6/Pi^2 = .607927101854...
%C A080058 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) = 3.66565771136..., where x=(1/zeta(2)) and m=floor(log(1-x)/log(x))=1.
%C A080058 See A077468 for Mathematica program by _Robert G. Wilson v_.
%F A080058 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(2)) and frac(y) = y - floor(y).
%e A080058 a(3)=8 since (1/zeta(2)) +(1/zeta(2))^2 +(1/zeta(2))^8 < 1 and (1/zeta(2)) +(1/zeta(2))^2 +(1/zeta(2))^k > 1 for 2<k<8.
%Y A080058 Cf. A077468, A080057, A080059, A229099.
%K A080058 nonn,changed
%O A080058 1,2
%A A080058 _Benoit Cloitre_ and _Paul D. Hanna_, Jan 23 2003