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
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
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