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A320309
Numerators of the fractions a(0)/(a(1) - a(0)), a(1)/(a(2) - a(1)), a(2)/(a(3) - a(2)), ... such that the sum 1/a(0) + Sum_{n>=1} a(n-1)/(a(n) - a(n-1)) has the concatenation of these numerators as decimal part. Case a(0) = 38.
15
38, 145, 78285806, 956422831811259761822, 37816270981051548637992987104427146105734001128733588866
OFFSET
0,1
COMMENTS
It appears that fractions of this kind exist only for a(0) equal to 3 (A320306), 10 (A320307), 13 (A320308) and 38 (this sequence).
Next term has 147 digits. - Giovanni Resta, Oct 11 2018
EXAMPLE
1/38 = 0.02631...
At the beginning instead of 38 we have 02 as first decimal digits. Adding the second term this is fixed.
1/38 + 38/(145 - 38) = 0.38145597...
1/38 + 38/(145 - 38) + 145/(78285806 - 145) = 0.3814559778285806137...
The sum is 0.38 145 78285806 956422831811259761822 ...
MAPLE
P:=proc(q, h) local a, b, d, n, t, x; x:=h+1; a:=1/h; b:=ilog10(h)+1;
d:=h; print(d); t:=1/a; for n from x to q do if
trunc(evalf(a+t/(n-t), 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:=n;
x:=n+1; print(n); fi; od; end: P(10^20, 38);
KEYWORD
nonn,base
AUTHOR
Paolo P. Lava, Oct 11 2018
EXTENSIONS
a(4)-a(5) from Giovanni Resta, Oct 11 2018
STATUS
approved