A320306 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) = 3.

3, 75, 86640, 9805882200056, 213947098435952365919629345554065819

Offset

0,1

Comments

It appears that fractions of this kind exist only for a(0) equal to 3 (this sequence), 10 (A320307), 13 (A320308) and 38 (A320309).

Next term has 93 digits. - Giovanni Resta, Oct 11 2018

Links

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

Example

1/3 = 0.3333...
1/3 + 3/(75 - 3) = 0.375000...
1/3 + 3/(75 - 3) + 75/(86640 - 75) = 0.37586640097...
The sum is 0.3 75 86640 9805882200056 ...

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, 3);

Crossrefs

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

Sequence in context: A300924 A183290 A336062 * A359500 A238000 A308866

Adjacent sequences: A320303 A320304 A320305 * A320307 A320308 A320309

Keyword

nonn,base

Author

Paolo P. Lava, Oct 11 2018

Extensions

a(4)-a(5) from Giovanni Resta, Oct 11 2018

Status

approved