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A305663
Denominators a(n) of the fractions Sum_{n>=1} {n/a(n)} = 1/a(1) + 2/a(2) + 3/a(3) + ... such that the sum has the concatenation of these denominators as decimal part. Case a(1) = 31.
26
31, 7, 975425, 85939741154, 936552965614980263201, 933486208332286775628057914016814688052592
OFFSET
1,1
COMMENTS
It appears that fractions of this kind exist only for a(1) equal to 3 (A304288), 10 (A304289), 11 (A305661), 14 (A305662), and 31 (this sequence).
For each of the other cases, the terms of the sequence are in increasing order, but this sequence begins with 31, 7, ... (see Example).
a(7) has 84 digits. - Giovanni Resta, Jun 08 2018
EXAMPLE
1/31 = 0.03225... At the beginning instead of 31 we have 03 as first decimal digits. Adding the second term this is fixed.
1/31 + 2/7 = 0.317972...
1/31 + 2/7 + 3/975425 = 0.317975425812...
The sum is 0.31 7 975425...
MAPLE
P:=proc(q, h) local a, b, d, n, t; a:=1/h; b:=ilog10(h)+1; d:=h; print(d);
t:=2; for n from 1 to q do if trunc(evalf(a+t/n, 100)*10^(b+ilog10(n)+1))=d*10^(ilog10(n)+1)+n then b:=b+ilog10(n)+1; d:=d*10^(ilog10(n)+1)+n; a:=a+t/n; t:=t+1; print(n); fi; od; end: P(10^20, 31);
KEYWORD
nonn,base
AUTHOR
Paolo P. Lava, Jun 08 2018
EXTENSIONS
a(4)-a(6) from Giovanni Resta, Jun 08 2018
STATUS
approved