Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #50 Jul 30 2021 13:24:36
%S 1,37,12345679,4115226337448559670781893,
%T 1371742112482853223593964334705075445816186556927297668038408779149519890260631
%N Ratios R(k)/k for which R(k) / k is an integer, where R(k) = A002275(k) is a repunit.
%C This is the sequence where fractions of repunits (A002275) and their number of digits in base 10 R(k) / k are integers, where k is A014950(n). This happens for all k of the form k=3^m; this is true because R(3k) / R(k) = 10^(2k) + 10^n*k + 1, which is divisible by 3. Therefore R(3^m) is divisible by 3^m by induction on m. There are additional solutions in A014950.
%F a(n) = A002275(A014950(n))/A014950(n).
%e For n = 2, a(2) = 111/3 = 37. For n = 3, a(3) = 111111111/9 = 12345679.
%t s = Join[{1}, Select[Range[3, 81, 6], PowerMod[10, #, #] == 1 &]]; Table[(10^n - 1)/(9*n), {n, s}] (* _Amiram Eldar_, Jun 20 2021 *)
%o (Python) [(10**n-1)//(9*n) for n in range(1, 300) if not (10**n-1)//9 % n]
%Y Cf. A002275, A014950.
%K base,nonn
%O 1,2
%A _Thomas T. Burgess_, Jun 20 2021