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A325726
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) = 10 and a(n) > a(n-1).
3
10, 321, 688679, 648399312079, 795011217364419041371938, 378887020322265448186183258957408252353018723756
OFFSET
1,1
COMMENTS
At any step only the least value greater than a(n) is taken into consideration. In fact, instead of 321, as a(2) we could choose 515, 1290, ...
LINKS
Eric Weisstein's World of Mathematics, Egyptian fraction
EXAMPLE
1/10 = 0.1000...
1/10 + 1/(321-10) = 0.10321543...
1/10 + 1/(321-10) + 1/(688679-321+10) = 0.10321688679494...
The sum is 0.10 321 688679 ...
MAPLE
P:=proc(q, h) local a, b, d, n, t, z; a:=1/h; b:=length(h); d:=h;
print(d); t:=h; for n from t+1 to q do
z:=evalf(evalf(a+1/(n-t), 100)*10^(b+length(n)), 100);
z:=trunc(z-frac(z)); if z=d*10^length(n)+n then b:=b+length(n);
d:=d*10^length(n)+n; t:=n-t; a:=a+1/t; print(n); fi; od; end:
P(10^20, 10);
KEYWORD
nonn,base
AUTHOR
Paolo P. Lava, May 17 2019
EXTENSIONS
a(4)-a(6) from Giovanni Resta, May 17 2019
STATUS
approved