cp's OEIS Frontend

A080057 Greedy powers of exp(-gamma): Sum_{n>=1} exp(-gamma)^a(n) = 1, where exp(-gamma) = exp(-.57721566490153286...) = .561459483566885169...

%I A080057 #11 Sep 01 2025 23:44:55
%S A080057 1,2,4,7,9,13,15,17,20,21,23,27,29,34,35,38,40,42,43,46,48,49,51,54,
%T A080057 57,58,61,64,65,68,73,74,80,83,85,87,89,98,100,101,104,105,107,110,
%U A080057 113,116,117,120,122,123,126,128,132,136,139,142,149,152,156,157,160,161,163
%N A080057 Greedy powers of exp(-gamma): Sum_{n>=1} exp(-gamma)^a(n) = 1, where exp(-gamma) = exp(-.57721566490153286...) = .561459483566885169...
%C A080057 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) = 2.909795625992782..., where x=exp(-gamma) and m=floor(log(1-x)/log(x))=1.
%C A080057 See A077468 for Mathematica program by _Robert G. Wilson v_.
%F A080057 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=(exp(-Gamma)) and frac(y) = y - floor(y).
%e A080057 a(3)=4 since exp(-gamma) + exp(-gamma)^2 + exp(-gamma)^4 < 1 and exp(-gamma) + exp(-gamma)^2 + exp(-gamma)^k > 1 for 2<k<4.
%Y A080057 Cf. A077468, A076802, A080056, A080058.
%K A080057 easy,nonn,changed
%O A080057 1,2
%A A080057 _Benoit Cloitre_ and _Paul D. Hanna_, Jan 23 2003