%I #15 Oct 19 2018 04:55:20

%S 10,3167,9115670120,542008360464753577575056956,

%T 7830689983639267579884170593492086524040157478312672952864203282974067

%N Numerators of the fractions a(0)/(a(1) - a(0)), a(1)/(a(2) - a(1)), a(2)/(a(3) - a(2)), ... such that the sum 1/a(0) + Sum_{n>=1} a(n-1)/(a(n) - a(n-1)) has the concatenation of these numerators as decimal part. Case a(0) = 10.

%C It appears that fractions of this kind exist only for a(0) equal to 3 (A320306), 10 (this sequence), 13 (A320308) and 38 (A320309).

%C Next term has 184 digits. - _Giovanni Resta_, Oct 11 2018

%e 1/10 = 0.1000...

%e 1/10 + 10/(3167 - 10) = 0.103167564...

%e 1/10 + 10/(3167 - 10) + 3167/(9115670120 - 3167) = 0.1031679115670120373...

%e The sum is 0.10 3167 9115670120 542008360464753577575056956 ...

%p P:=proc(q,h) local a,b,d,n,t,x; x:=h+1; a:=1/h; b:=ilog10(h)+1;

%p d:=h; print(d); t:=1/a; for n from x to q do if

%p trunc(evalf(a+t/(n-t),100)*10^(b+ilog10(n)+1))=d*10^(ilog10(n)+1)+n

%p then b:=b+ilog10(n)+1; d:=d*10^(ilog10(n)+1)+n; a:=a+t/(n-t); t:=n;

%p x:=n+1; print(n); fi; od; end: P(10^20,10);

%Y Cf. A302932, A302933, A303388, A304285, A304286, A304287, A304288, A304289, A305661, A305662, A305663, A305664, A305665, A305666, A320306, A320308, A320309.

%K nonn,base

%O 0,1

%A _Paolo P. Lava_, Oct 11 2018

%E a(3)-a(5) from _Giovanni Resta_, Oct 11 2018