A320308 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.

13, 247, 890970451, 540818819108243685091145, 77126435926826212355124730859827553231849402446824773070904238

Offset

0,1

Comments

It appears that fractions of this kind exist only for a(0) equal to 3 (A320306), 10 (A320307), 13 (this sequence) and 38 (A320309).

Next term has 163 digits. - Giovanni Resta, Oct 11 2018

Links

Table of n, a(n) for n=0..4.

Example

1/13 = 0.07692...
At the beginning instead of 13 we have 07 as first decimal digits. Adding the second term this is fixed.
1/13 + 13/(247 - 13) = 0.13247863...
1/13 + 13/(247 - 13) + 247/(890970451 - 247) = 0.13247890970451376...
The sum is 0.13 247 890970451 540818819108243685091145 ...

Maple

P:=proc(q, h) local a, b, d, n, t, x; x:=h+1; a:=1/h; b:=ilog10(h)+1;
d:=h; print(d); t:=1/a; for n from x 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(10^20, 13);

Crossrefs

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

Sequence in context: A196665 A027400 A053100 * A219059 A218315 A183416

Adjacent sequences: A320305 A320306 A320307 * A320309 A320310 A320311

Keyword

nonn,base

Author

Paolo P. Lava, Oct 11 2018

Extensions

a(3)-a(5) from Giovanni Resta, Oct 11 2018

Status

approved