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

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, 52, 55, 56, 58, 59, 61, 63, 64, 71, 72, 78, 83, 84, 85, 88, 89, 93, 94, 96, 98, 100, 101, 104, 105, 106, 109, 114, 116, 117, 120, 121, 122, 125, 133, 134, 138, 140, 141, 142

Offset: 1

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Keywords

• easy
• nonn

Comments

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.

Examples

			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).
		

Crossrefs

Cf. A077468-A077475.

Programs

Maple

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;

Mathematica

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 *)

Formula

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).

Extensions

Corrected by T. D. Noe, Nov 02 2006