A320309 - OEIS

%I #15 Oct 19 2018 04:54:38

%S 38,145,78285806,956422831811259761822,

%T 37816270981051548637992987104427146105734001128733588866

%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) = 38.

%C It appears that fractions of this kind exist only for a(0) equal to 3 (A320306), 10 (A320307), 13 (A320308) and 38 (this sequence).

%C Next term has 147 digits. - _Giovanni Resta_, Oct 11 2018

%e 1/38 = 0.02631...

%e At the beginning instead of 38 we have 02 as first decimal digits. Adding the second term this is fixed.

%e 1/38 + 38/(145 - 38) = 0.38145597...

%e 1/38 + 38/(145 - 38) + 145/(78285806 - 145) = 0.3814559778285806137...

%e The sum is 0.38 145 78285806 956422831811259761822 ...

%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(10^20,38);

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

%K nonn,base

%O 0,1

%A _Paolo P. Lava_, Oct 11 2018

%E a(4)-a(5) from _Giovanni Resta_, Oct 11 2018