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A345467 Ratios R(k)/k for which R(k) / k is an integer, where R(k) = A002275(k) is a repunit. 0
1, 37, 12345679, 4115226337448559670781893, 1371742112482853223593964334705075445816186556927297668038408779149519890260631 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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.
LINKS
FORMULA
a(n) = A002275(A014950(n))/A014950(n).
EXAMPLE
For n = 2, a(2) = 111/3 = 37. For n = 3, a(3) = 111111111/9 = 12345679.
MATHEMATICA
s = Join[{1}, Select[Range[3, 81, 6], PowerMod[10, #, #] == 1 &]]; Table[(10^n - 1)/(9*n), {n, s}] (* Amiram Eldar, Jun 20 2021 *)
PROG
(Python) [(10**n-1)//(9*n) for n in range(1, 300) if not (10**n-1)//9 % n]
CROSSREFS
Sequence in context: A289820 A087020 A238840 * A329226 A171407 A023930
KEYWORD
base,nonn
AUTHOR
Thomas T. Burgess, Jun 20 2021
STATUS
approved

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Last modified July 24 20:07 EDT 2024. Contains 374585 sequences. (Running on oeis4.)