%I #9 Oct 19 2018 04:54:47

%S 1,13,276,69578731,8400530190113978524440,

%T 49897684059619962746027095165660646312316539507342111435767

%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 Sum_{n>=1} a(n-1)/(a(n) - a(n-1)) has the concatenation of these numerators, starting from a(1), as decimal part. Case a(0) = 1, a(1) = 13.

%C It appears that fractions of this kind with a(0)=1 exist only for a(1) equal to 4 (A320335) and 13 (this sequence).

%C Next term has 153 digits. - _Giovanni Resta_, Oct 11 2018

%e 1/(13-1) = 0.0833...

%e At the beginning instead of 13 we have 08 as first decimal digit. Adding the second term this is fixed.

%e 1/(13-1) + 13/(276 - 13) = 0.13276299...

%e 1/(13-1) + 13/(276 - 13) + 276/(69578731 - 276) = 0.1327669578731757 ...

%e The sum is 0.13 276 69578731 8400530190113978524440 ...

%p P:=proc(q,h) local a, b, d,t, x,n; x:=1; a:=1/(h-1); b:=ilog10(h-1)+1; d:=h; print(d); t:=h; for n from h+1 to q do if trunc(evalf(a+t/(n-t),100)*10^(b+ilog10(n)+1))=d*10^(ilog10(n)+1)+n then b:=b+ilog10(n)+1; d:=d*10^(ilog10(n)+1)+n; a:=a+t/(n-t); t:=n; x:=n+1; print(n); fi; od; end: P(10^10,13);

%Y Cf. A302932, A302933, A303388, A304285, A304286, A304287, A304288, A304289, A305661, A305662, A305663, A305664, A305665, A305666, A320306, A320307, A320308, A320309, A320335.

%K nonn,base

%O 0,2

%A _Paolo P. Lava_, Oct 11 2018

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