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%I #14 Jun 07 2019 08:57:21
%S 3,53,5254,942711270,60527555104759884,
%T 6079815437830353357655363418656533,
%U 36605822957679968262595918562001251109330115418597396926274122249725
%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) = 3 and a(n) > a(n-1).
%C At any step only the least value greater than a(n) is taken into consideration. In fact, instead of 53, as a(2) we could choose 76, 367, 3366, 3666, 33367, 34350, 333366, ...
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EgyptianFraction.html">Egyptian fraction</a>
%e 1/3 = 0.3333...
%e 1/3 + 1/(53-3) = 0.353333...
%e 1/3 + 1/(53-3) + 1/(5254-53+3) = 0.3535254932...
%e The sum is 0.3 53 5254 ...
%p P:=proc(q, h) local a, b, d, n, t, z; a:=1/h; b:=length(h);d:=h; print(d); t:=h;
%p for n from t+1 to q do 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(10^20, 3);
%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, A325727, A325728.
%K nonn,base
%O 1,1
%A _Paolo P. Lava_, May 17 2019
%E a(4) - a(7) from _Giovanni Resta_, May 17 2019
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