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%I #17 Jan 05 2020 07:18:59
%S 10,100,316,5169,183766,972915,8110039
%N Engel expansion whose sum has the concatenation of its terms as decimal part. Case a(1) = 10.
%C 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.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EngelExpansion.html ">Engel expansion</a>
%e 1/10 = 0.10000...
%e 1/10 + 1/(10*100) = 0.10100000...
%e 1/10 + 1/(10*100) + 1/(10*100*316) = 0.10100316455...
%e The sum is 0.10 100 316 5169 ...
%p P:=proc(q, h) local a, b,c, d, n, x; x:=1; a:=1/h; b:=ilog10(h)+1;
%p 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
%p then x:=n+1; b:=b+ilog10(n)+1; d:=d*10^(ilog10(n)+1)+n;
%p a:=a+1/(c*n); c:=c*n; print(n); fi; od; end: P(10^9,10);
%Y Cf. A302932, A302933, A303388, A304285, A304286, A304287, A304288, A304289, A305661, A305662, A305663, A305664, A305665, A305666, A305667.
%K nonn,base,fini,full
%O 1,1
%A _Paolo P. Lava_, Jun 12 2018
%E a(5)-a(7) from _Giovanni Resta_, Jun 12 2018
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