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A345467
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Ratios R(k)/k for which R(k) / k is an integer, where R(k) = A002275(k) is a repunit.
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0
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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For n = 2, a(2) = 111/3 = 37. For n = 3, a(3) = 111111111/9 = 12345679.
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MATHEMATICA
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s = Join[{1}, Select[Range[3, 81, 6], PowerMod[10, #, #] == 1 &]]; Table[(10^n - 1)/(9*n), {n, s}] (* Amiram Eldar, Jun 20 2021 *)
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PROG
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(Python) [(10**n-1)//(9*n) for n in range(1, 300) if not (10**n-1)//9 % n]
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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STATUS
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approved
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