A307007 - OEIS

%I #24 Mar 26 2019 16:14:52

%S 3,44,70171,99999262192,91098508760349172092,

%T 970792725489545464249914539975116316038,

%U 931700887896779243871964259462997210573060273337039138324846507043947496698605

%N a(n) is defined by the condition that the decimal expansion of the Sum_{n>=1} 1/(n*a(n)) = 1/(1*a(1)) + 1/(2*a(2)) + 1/(3*a(3)) + ... begins with the concatenation of these numbers; also a(1) = 3 and a(n) > a(n-1).

%C If we had Sum_{n>=1} (-1)^(n-1)/(n*a(n)) = 1/(1*a(1)) - 1/(2*a(2)) + 1/(3*a(3)) - ... the terms would be only 3, 317, 61469 because we get 0.3317614690547... and the zero after 61469 cannot be covered by any number.

%C At any step only the least value greater than a(n) is taken into consideration. As a(2) we could choose 44, 347, 3348, 33348, ..., 3...348. Again, if a(2) = 44 then we could choose as a(3) 70171, 697447, ...

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EgyptianFraction.html">Egyptian fraction</a>

%e 1/(1*3) = 0.3333...

%e 1/(1*3) + 1/(2*44) = 0.344696...

%e 1/(1*3) + 1/(2*44) + 1/(3*70171) = 0.34470171999...

%e The sum is 0.3 44 70171 99999262192 ...

%p P:=proc(q, h) local a,b,d,n,t,z; a:=1/h; b:=ilog10(h)+1;

%p d:=h; print(d); t:=2; for n from 1 to q do

%p z:=evalf(evalf(a+1/(t*n),100)*10^(b+ilog10(n)+1),100);

%p z:=trunc(z-frac(z)); if z=d*10^(ilog10(n)+1)+n

%p then b:=b+ilog10(n)+1; d:=d*10^(ilog10(n)+1)+n; a:=a+1/(t*n); t:=t+1;

%p print(n); fi; od; end: P(10^9,3);

%Y Cf. A304288, A304289, A305661, A305662, A305663, A305664, A305665, A305666, A305667, A305668, A320023, A320284, A320306, A320307, A320308, A320309, A320335, A320336, A324222, A324223.

%K nonn,more

%O 1,1

%A _Paolo P. Lava_, Mar 19 2019

%E a(3)-a(7) from _Giovanni Resta_, Mar 19 2019