A305666 Denominators of the fraction Sum_{n>=0} {a(n)/a(n+1)} = a(0)/a(1) + a(1)/a(2) + a(2)/a(3) + ... such that the sum has the concatenation of these denominators as decimal part. Case a(0) = 1 and a(1) = 10.

1, 10, 3162, 9048102540, 9979213683484174723652691445

Offset

0,2

Comments

It appears that there are only three sequences of this kind, with a(1) = 3 (A305664), a(1) = 4 (A305665) and a(1) = 10 (this sequence).

a(5) has 73 digits. - Giovanni Resta, Jun 08 2018

Links

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

Example

1/10 = 0.1000...
1/10 + 10/3162 = 0.103162555...
1/10 + 10/3162 + 3162/9048102540 = 0.1031629048102540988...
The sum is 0.10 3162 9048102540 ...

Maple

P:=proc(q, h) local a, b, d, n, t, x; x:=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, 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:=n; x:=n+1; print(n); fi; od; end: P(10^20, 10);

Crossrefs

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

Sequence in context: A208185 A123377 A061543 * A320307 A225764 A243008

Adjacent sequences: A305663 A305664 A305665 * A305667 A305668 A305669

Keyword

nonn,base

Author

Paolo P. Lava, Jun 08 2018

Extensions

a(3)-a(4) from Giovanni Resta, Jun 08 2018

a(0)=1 inserted. - R. J. Mathar, Jun 19 2021

Status

approved