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%I #15 Oct 19 2018 04:55:05
%S 13,247,890970451,540818819108243685091145,
%T 77126435926826212355124730859827553231849402446824773070904238
%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) = 13.
%C It appears that fractions of this kind exist only for a(0) equal to 3 (A320306), 10 (A320307), 13 (this sequence) and 38 (A320309).
%C Next term has 163 digits. - _Giovanni Resta_, Oct 11 2018
%e 1/13 = 0.07692...
%e At the beginning instead of 13 we have 07 as first decimal digits. Adding the second term this is fixed.
%e 1/13 + 13/(247 - 13) = 0.13247863...
%e 1/13 + 13/(247 - 13) + 247/(890970451 - 247) = 0.13247890970451376...
%e The sum is 0.13 247 890970451 540818819108243685091145 ...
%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(10^20,13);
%Y Cf. A302932, A302933, A303388, A304285, A304286, A304287, A304288, A304289, A305661, A305662, A305663, A305664, A305665, A305666, A320306, A320307, 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
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