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A365933 a(n) is the period of the remainders when repdigits are divided by n.

%I A365933 #74 Jan 23 2024 13:25:27
%S A365933 1,9,27,9,9,27,54,9,81,9,18,27,54,54,27,9,144,81,162,9,54,18,198,27,9,
%T A365933 54,243,54,252,27,135,9,54,144,54,81,27,162,54,9,45,54,189,18,81,198,
%U A365933 414,27,378,9,432,54,117,243,18,54,162,252,522,27,540,135,162,9,54
%N A365933 a(n) is the period of the remainders when repdigits are divided by n.
%C A365933 For n>1: Periods are divisible by 9 (= a full cycle in the sequence of repdigits). a(n)/9 is the period of the remainders when repunits are divided by n. So the digit part of the repdigits has no effect on periods generally. For most n the beginning of the periodic part is always A010785(1). If n is a term of A083118 the periodic part starts later after some initial remainders that do not repeat.
%H A365933 Karl-Heinz Hofmann, <a href="/A365933/a365933_3.txt">Table with additional information</a>.
%e A365933 For n = 6:                Remainders of A010785(1..54) mod n.
%e A365933 A010785( 1...9) mod n:      [1, 2, 3, 4, 5, 0, 1, 2, 3]
%e A365933 A010785(10..18) mod n:      [5, 4, 3, 2, 1, 0, 5, 4, 3]
%e A365933 A010785(19..27) mod n:      [3, 0, 3, 0, 3, 0, 3, 0, 3]
%e A365933 So the period is 3*9 = 27. Thus a(n) = 27. And the pattern seen above starts again:
%e A365933 A010785(28..36) mod n:      [1, 2, 3, 4, 5, 0, 1, 2, 3]
%e A365933 A010785(37..45) mod n:      [5, 4, 3, 2, 1, 0, 5, 4, 3]
%e A365933 A010785(46..54) mod n:      [3, 0, 3, 0, 3, 0, 3, 0, 3]
%o A365933 (Python)
%o A365933 def A365933(n):
%o A365933     if n == 1: return 1
%o A365933     remainders, exponent = [], 1
%o A365933     while (rem:=(10**exponent // 9 % n)) not in remainders:
%o A365933         remainders.append(rem); exponent += 1
%o A365933     return (exponent - remainders.index(rem) - 1) * 9
%o A365933 (Python)
%o A365933 def A365933(n):
%o A365933     if n==1: return 1
%o A365933     a,b,x,y=1,1,1%n,11%n
%o A365933     while x!=y:
%o A365933         if a==b:
%o A365933             a<<=1
%o A365933             x,b=y,0
%o A365933         y = (10*y+1)%n
%o A365933         b+=1
%o A365933     return 9*b # _Chai Wah Wu_, Jan 23 2024
%Y A365933 Cf. A010785, A002275.
%Y A365933 Cf. A305322 (divisor 3), A002279 (divisor 5), A366596 (divisor 7).
%Y A365933 Cf. A083118 (the impossible divisors).
%K A365933 nonn,base
%O A365933 1,2
%A A365933 _Karl-Heinz Hofmann_, Nov 07 2023