A345467 - OEIS

A345467

Ratios R(k)/k for which R(k) / k is an integer, where R(k) = A002275(k) is a repunit.

1, 37, 12345679, 4115226337448559670781893, 1371742112482853223593964334705075445816186556927297668038408779149519890260631

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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

Table of n, a(n) for n=1..5.

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

Cf. A002275, A014950.