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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) = 14 and a(n) > a(n-1).
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%I #11 Jun 07 2019 08:57:58

%S 14,28,71192,3951239529,91599717337708557840,

%T 25287155431709811559363106042646332272281

%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) = 14 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 1/14 = 0.071428...

%e 1/14 + 1/(28-14) = 0.1428571...

%e 1/14 + 1/(28-14) + 1/(71192-28+14) = 0.142871192142...

%e The sum is 0.14 28 71192...

%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, 14);

%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, A325725, A325726, A325727.

%K nonn,base

%O 1,1

%A _Paolo P. Lava_, May 17 2019

%E a(4)-a(6) from _Giovanni Resta_, May 17 2019