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).

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

Table of n, a(n) for n=1..7.

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);

Crossrefs

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

Sequence in context: A369944 A193623 A102811 * A142600 A212999 A103980

Adjacent sequences: A307004 A307005 A307006 * A307008 A307009 A307010

Keyword

nonn,more

Author

Paolo P. Lava, Mar 19 2019

Extensions

a(3)-a(7) from Giovanni Resta, Mar 19 2019

Status

approved