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A307007
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).
7
3, 44, 70171, 99999262192, 91098508760349172092, 970792725489545464249914539975116316038, 931700887896779243871964259462997210573060273337039138324846507043947496698605
OFFSET
1,1
COMMENTS
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.
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, ...
LINKS
Eric Weisstein's World of Mathematics, Egyptian fraction
EXAMPLE
1/(1*3) = 0.3333...
1/(1*3) + 1/(2*44) = 0.344696...
1/(1*3) + 1/(2*44) + 1/(3*70171) = 0.34470171999...
The sum is 0.3 44 70171 99999262192 ...
MAPLE
P:=proc(q, h) local a, b, d, n, t, z; a:=1/h; b:=ilog10(h)+1;
d:=h; print(d); t:=2; for n from 1 to q do
z:=evalf(evalf(a+1/(t*n), 100)*10^(b+ilog10(n)+1), 100);
z:=trunc(z-frac(z)); if z=d*10^(ilog10(n)+1)+n
then b:=b+ilog10(n)+1; d:=d*10^(ilog10(n)+1)+n; a:=a+1/(t*n); t:=t+1;
print(n); fi; od; end: P(10^9, 3);
KEYWORD
nonn,more
AUTHOR
Paolo P. Lava, Mar 19 2019
EXTENSIONS
a(3)-a(7) from Giovanni Resta, Mar 19 2019
STATUS
approved