A112595 Sequence of numerators of the continued fraction derived from the sequence of the number of distinct factors of a number (A001221, also called omega(n)).

0, 1, 1, 2, 3, 8, 11, 19, 30, 79, 109, 297, 406, 1109, 2624, 3733, 6357, 16447, 22804, 62055, 146914, 355883, 502797, 1361477, 1864274, 5090025, 6954299, 18998623, 25952922, 96857389, 122810311, 219667700, 562145711, 1343959122, 3250063955

Offset: 1

Author

Giorgio Balzarotti and Paolo P. Lava, Dec 19 2005

Keywords

• frac
• nonn

Comments

The limits of the continued fraction is Cd = 0.6123687534182316423985073896748729172179677660718454489694806870..., i.e. the number associated to the sequence of number of distinct primes dividing n.

Examples

			a[1]=d[1]=0 (d[1] is the first element of A001221, i.e. the number of distinct primes dividing 1).
a[2]=d[2]*a[1]+1=0*1+1=1;
a[3]=d[3]*a[2]+a[1]=1*1+0=1.
		

Crossrefs

Cf. A001221, A112596.

Programs

Maple

a:=proc(N) # A is numerator of the continued fraction # B is denominator of the continued fraction # d is the sequence of the number of divisors of a number (A001221), d[1] is the first element. A[1]:=d[1]; A[2]:=d[2]*A[1]+1; B[1]:=1; B[2]:=d[2]*B[1]; for n from 2 by 1 to N-1 do A[n+1]:=d[n+1]*A[n]+A[n-1]; B[n+1]:=d[n+1]*B[n]+B[n-1]; od; end: