A076796 Greedy powers of Pi/4: Sum_{n>=1} (Pi/4)^a(n) = 1.

1, 7, 15, 24, 32, 39, 47, 59, 79, 88, 102, 111, 134, 148, 158, 164, 172, 190, 206, 214, 220, 233, 241, 251, 263, 271, 283, 292, 307, 314, 322, 329, 337, 350, 358, 364, 373, 384, 399, 413, 438, 446, 456, 462, 475, 481, 494, 502, 516, 529, 536, 552, 559, 567

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

			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.
		

Crossrefs

Cf. A077468 - A077475.

Programs

Maple

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;

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

Extensions

Typos in data corrected by Sean A. Irvine, Apr 16 2025