OFFSET
0,1
COMMENTS
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 unit fraction series - they are dense on the interval (2,oo).
EXAMPLE
The partial continued fraction must always be strictly larger than the partial unit fractions:
[1] cannot be since 1 = 1.
[2] can be since 2 > 1/2.
[2,1] can be since 2+1/1 > 1/2+1/1.
[2,1,1] cannot be since 2+1/(1+1/1) = 1/2+1/1+1/1.
[2,1,2] can be since 2+1/(1+1/2) > 1/2+1/1+1/2.
...
sum(1/a[n]) = 2.71053359137351078733864566... (A364873).
PROG
(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};
CROSSREFS
KEYWORD
nonn,cofr
AUTHOR
Rok Cestnik, Aug 11 2023
STATUS
approved