A079938 Greedy frac multiples of Pi: a(1)=1, Sum_{n>=1} frac(a(n)*Pi) = 1.

1, 2, 3, 8, 99, 33102, 66317, 265381, 1360120, 25510582, 78256779, 156513558, 209259755, 340262731, 1963319607, 6701487259, 8664806866, 13402974518, 20104461777, 26805949036, 33507436295, 40208923554, 46910410813

Offset: 1

Author

Benoit Cloitre and Paul D. Hanna, Jan 21 2003

Keywords

• nonn
• changed

Comments

The n-th greedy frac multiple of x is the smallest integer that does not cause Sum_{k=1..n} frac(a(k)*x) to exceed unity; an infinite number of terms appear as the denominators of the convergents to the continued fraction of x.

Examples

			a(4) = 8 since frac(1x*) + frac(2*x) + frac(3*x) + frac(8*x) < 1, while frac(1*x) + frac(2*x) + frac(3*x) + frac(k*x) > 1 for all k > 3 and k < 8.
		

Crossrefs

Cf. A002486 (denominators of convergents to Pi), A079934, A079937, A079939.

Programs

Maple

Digits := 100: a := []: s := 0: x := Pi: for n from 1 to 10000000 do: temp := evalf(s+frac(n*x)): if (temp<1.0) then a := [op(a),n]: print(n): s := s+evalf(frac(n*x)): fi: od: a;

PARI

first(n)=my(v=vector(n),s=1.,p=Pi-3,k); for(m=1,oo, my(t=frac(p*m)); if(tCharles R Greathouse IV, Jul 25 2024

Extensions

a(9) from Mark Hudson, Jan 30 2003

a(10)-a(23) from Charles R Greathouse IV, Jul 26 2024