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A320306
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) = 3.
15
3, 75, 86640, 9805882200056, 213947098435952365919629345554065819
OFFSET
0,1
COMMENTS
It appears that fractions of this kind exist only for a(0) equal to 3 (this sequence), 10 (A320307), 13 (A320308) and 38 (A320309).
Next term has 93 digits. - Giovanni Resta, Oct 11 2018
EXAMPLE
1/3 = 0.3333...
1/3 + 3/(75 - 3) = 0.375000...
1/3 + 3/(75 - 3) + 75/(86640 - 75) = 0.37586640097...
The sum is 0.3 75 86640 9805882200056 ...
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, 3);
KEYWORD
nonn,base
AUTHOR
Paolo P. Lava, Oct 11 2018
EXTENSIONS
a(4)-a(5) from Giovanni Resta, Oct 11 2018
STATUS
approved