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A076798 Greedy powers of (Pi/6): Sum_{n>=1} (Pi/6)^a(n) = 1.

%I A076798 #9 Jul 08 2017 12:46:56
%S A076798 1,2,3,5,7,8,10,11,12,14,20,21,22,25,28,33,35,37,38,39,44,45,47,49,50,
%T A076798 52,55,56,58,59,61,63,64,71,72,78,83,84,85,88,89,93,94,96,98,100,101,
%U A076798 104,105,106,109,114,116,117,120,121,122,125,133,134,138,140,141,142
%N A076798 Greedy powers of (Pi/6): Sum_{n>=1} (Pi/6)^a(n) = 1.
%C A076798 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.
%F A076798 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=(pi/6) and frac(y) = y - floor(y).
%e A076798 a(4)=5 since (Pi/6) +(Pi/6)^2 +(Pi/6)^3 +(Pi/6)^5 < 1 and (Pi/6) +(Pi/6)^2 +(Pi/6)^3 +(Pi/6)^4 > 1; since the power 4 makes the sum > 1, then 5 is the 4th greedy power of (Pi/6).
%p A076798 Digits := 400: summe := 0.0: p := evalf(Pi / 6.): pexp := p: a := []: for i from 1 to 800 do: if summe + pexp < 1 then a := [op(a),i]: summe := summe + pexp: fi: pexp := pexp * p: od: a;
%t A076798 g[1] = 1; g[n_] := g[n] = With[{x = Pi/6}, Log[x, x^FractionalPart[g[n-1]] - x]]; a[n_] := Sum[Floor[g[k]], {k, 1, n}]; Table[a[n], {n, 1, 64}] (* _Jean-François Alcover_, Jul 08 2017 *)
%Y A076798 Cf. A077468-A077475.
%K A076798 easy,nonn
%O A076798 1,2
%A A076798 Ulrich Schimke (ulrschimke(AT)aol.com)
%E A076798 Corrected by _T. D. Noe_, Nov 02 2006