A364873 Decimal expansion of the lexicographically earliest continued fraction which equals its own sum of reciprocals.

2, 7, 1, 0, 5, 3, 3, 5, 9, 1, 3, 7, 3, 5, 1, 0, 7, 8, 7, 3, 3, 8, 6, 4, 5, 6, 6, 2, 0, 4, 8, 1, 7, 0, 1, 1, 1, 5, 1, 8, 3, 3, 4, 9, 9, 3, 0, 7, 0, 4, 4, 7, 6, 3, 7, 9, 4, 3, 4, 3, 9, 0, 9, 5, 0, 8, 3, 0, 4, 7, 0, 0, 0, 8, 2, 0, 7, 6, 8, 6, 1, 8, 7, 3, 1, 3, 1, 8, 2, 2, 1, 9, 6, 8, 7, 2, 2

Offset: 1

Author

Rok Cestnik, Aug 11 2023

Keywords

• nonn
• cons

Comments

This continued fraction (A364872) is the earliest infinite sequence {a0,a1,a2,a3,...} such that: a0+1/(a1+1/(a2+1/(a3+...))) = 1/a0 + 1/a1 + 1/a2 + 1/a3 + ....

There are infinitely many real numbers whose continued fraction is also their sum of reciprocals - they are dense on the interval (2,oo).

Examples

			2.71053359137351078733864566...
		

Crossrefs

Cf. A364872.

Programs

PARI

cf(a) = my(m=contfracpnqn(a)); m[1, 1]/m[2, 1];
uf(a) = sum(i=1, #a, 1/a[i]);
A364872(N) = {a=[2]; for(i=2, N, a=concat(a, if(cf(a)==uf(a), a[i-1], ceil(1/(cf(a)-uf(a))))); while(cf(a)<=uf(a), a[i]++)); a};
A364873(N) = {t=2; while(floor(10^N*cf(A364872(t))) != floor(10^N*cf(A364872(t+1))), t++); digits(floor(10^(N-1)*cf(A364872(t))))};