A305668 Engel expansion whose sum has the concatenation of its terms as decimal part. Case a(1) = 10.

10, 100, 316, 5169, 183766, 972915, 8110039

Offset

1,1

Comments

a(7) is the last term because the sequence cannot be extended. At any step a(n) is chosen as the least number greater than a(n-1) that meets the requirement. Up to 8110039 the sum is 0.10 100 316 5169 183766 972915 8110039 008537... but the next term would be less than 1/(10*100*316*5169*183766*972915*8110039^2) = 0.00 000 000 0000 000000 000000 00000000 005206195... and the zeros after 8110039 cannot be removed.

Links

Table of n, a(n) for n=1..7.

Eric Weisstein's World of Mathematics, Engel expansion

Example

1/10 = 0.10000...
1/10 + 1/(10*100) = 0.10100000...
1/10 + 1/(10*100) + 1/(10*100*316) = 0.10100316455...
The sum is 0.10 100 316 5169 ...

Maple

P:=proc(q, h) local a, b, c, d, n, x; x:=1; a:=1/h; b:=ilog10(h)+1;
c:=h; d:=h; print(d); for n from x to q do if trunc(evalf(a+1/(c*n), 100)*10^(b+ilog10(n)+1))=d*10^(ilog10(n)+1)+n
then x:=n+1; b:=b+ilog10(n)+1; d:=d*10^(ilog10(n)+1)+n;
a:=a+1/(c*n); c:=c*n; print(n); fi; od; end: P(10^9, 10);

Crossrefs

Cf. A302932, A302933, A303388, A304285, A304286, A304287, A304288, A304289, A305661, A305662, A305663, A305664, A305665, A305666, A305667.

Sequence in context: A207770 A207897 A207657 * A262761 A134556 A207449

Adjacent sequences: A305665 A305666 A305667 * A305669 A305670 A305671

Keyword

nonn,base,fini,full

Author

Paolo P. Lava, Jun 12 2018

Extensions

a(5)-a(7) from Giovanni Resta, Jun 12 2018

Status

approved