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%I #12 Jun 07 2019 08:57:45
%S 11,52,9943537,9881972526746,9201489001757012121335125,
%T 82921502495183923916318126922414429034029157972857
%N a(n) is defined by the condition that the decimal expansion of the Sum_{n>=1} 1/(Sum_{k=1..n} a(k)) = 1/a(1) + 1/(a(2)-a(1)) + 1/(a(3)-a(2)+a(1)) + ... begins with the concatenation of these numbers; also a(1) = 11 and a(n) > a(n-1).
%C At any step only the least value greater than a(n) is taken into consideration.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EgyptianFraction.html">Egyptian fraction</a>
%e 1/11 = 0.090909...
%e 1/11 + 1/(52-11) = 0.1152993...
%e 1/11 + 1/(52-11) + 1/(9943537-52+11) = 0.11529943537979...
%e The sum is 0.11 53 5254 ...
%p P:=proc(q, h) local a, b, d, n, t, z; a:=1/h; b:=length(h); d:=h;
%p print(d); t:=h; for n from t+1 to q do
%p z:=evalf(evalf(a+1/(n-t),100)*10^(b+length(n)), 100);
%p z:=trunc(z-frac(z)); if z=d*10^length(n)+n then b:=b+length(n);
%p d:=d*10^length(n)+n; t:=n-t; a:=a+1/t; print(n); fi; od; end:
%p P(10^20, 11);
%Y Cf. A304288, A304289, A305661, A305662, A305663, A305664, A305665, A305666, A305667, A305668, A307007, A307020, A307021, A307022, A320023, A320284, A320306, A320307, A320308, A320309, A320335, A320336, A324222, A324223, A325726, A325726, A325728.
%K nonn,base
%O 1,1
%A _Paolo P. Lava_, May 17 2019
%E a(4)-a(6) from _Giovanni Resta_, May 17 2019
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