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Greedy powers of Pi/4: Sum_{n>=1} (Pi/4)^a(n) = 1.
4

%I #4 Apr 11 2021 16:13:02

%S 1,7,15,24,32,39,47,59,79,88,102,111,134,148,158,164,172,190,206,214,

%T 220,233,24,1,251,263,271,283,292,307,314,322,329,337,350,358,364,373,

%U 384,399,413,438,446,456,462,475,481,494,502,51,6,529,536,552,559,567

%N Greedy powers of Pi/4: Sum_{n>=1} (Pi/4)^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.

%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=Pi/4 and frac(y) = y - floor(y).

%e Pi/4 + (Pi/4)^7 + (Pi/4)^15 < 1 and Pi/4 + (Pi/4)^7 + (Pi/4)^14 > 1; since the power 14 makes the sum > 1, 15 is the 3rd greedy power of Pi/4, so a(3)=15.

%p Digits := 400: summe := 0.0: p := evalf(Pi / 4.): 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;

%Y Cf. A077468 - A077475.

%K easy,nonn

%O 1,2

%A Ulrich Schimke (ulrschimke(AT)aol.com)