A080056 - OEIS

%I #8 Oct 31 2013 12:23:01

%S 1,3,5,16,22,24,28,34,37,43,45,49,51,54,57,59,65,68,70,74,80,88,94,97,

%T 100,103,108,111,113,116,122,127,129,132,137,141,143,148,151,156,161,

%U 164,166,172,174,177,184,189,202,204,208,213,216,219,225,227,238,247

%N Greedy powers of (2/Pi): sum_{n=1..inf} (2/Pi)^a(n) = 1.

%C 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..inf} log(1 + x^n)/log(x) = 4.2164448079..., where x=(2/Pi) and m=floor(log(1-x)/log(x))=2.

%F 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=(2/Pi) and frac(y) = y - floor(y). See A077468 for mathematica program by _Robert G. Wilson v_.

%e a(3)=5 since (2/Pi) +(2/Pi)^3 +(2/Pi)^5 < 1 and (2/Pi) +(2/Pi)^3 +(2/Pi)^k > 1 for 3<k<5.

%Y Cf. A077468, A080055, A080057.

%K easy,nonn

%O 1,2

%A _Benoit Cloitre_ and _Paul D. Hanna_, Jan 23 2003