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A350220 Composite numbers d such that the period k of the repetend of 1/d is > 1 and divides d-1, and d is the first such composite with a given period.

%I A350220 #31 May 21 2022 08:41:54
%S A350220 33,91,148,246,451,496,505,561,657,703,1035,1105,1912,2120,2465,2556,
%T A350220 2752,2821,4005,4141,5461,6525,6533,6565,6601,6700,7107,8695,8905,
%U A350220 8911,10585,11649,12403,12801,13366,13695,13833,14701,15211,15841,17120,18336,19345,19503,19900
%N A350220 Composite numbers d such that the period k of the repetend of 1/d is > 1 and divides d-1, and d is the first such composite with a given period.
%C A350220 This is a subset of sequence A351396 with the extra condition that d is included if and only if it is the smallest value of d with a given period. Thus, 246 is included because its period is 5 (repetend is 04065) and it is the first valid of d with this period and, moreover, 5 divides evenly into 245. However, 55 (which is in A351396) is excluded because although its period (2 based on a repetend of 18 for 1/55) divides evenly into 54, there is a smaller value of d (33) with this property and a period of 2 (1/33 has a repetend of 03).
%H A350220 Chai Wah Wu, <a href="/A350220/b350220.txt">Table of n, a(n) for n = 1..3839</a>
%H A350220 Barry Smyth, <a href="https://towardsdatascience.com/are-pseudoprimes-hiding-out-among-the-composite-reciprocals-c8952cfe1ab4">Are pseudoprimes hiding out among the composite reciprocals?</a>, Towards Data Science, Mar 25 2022.
%e A350220 33 is a term since 1/33 = 0.030303..., its repetend is 03, so its period is 2, 2 divides into 33-1 evenly, and there is no smaller value of d with this period.
%e A350220 91 is a term since 1/91 = 0.010989010989..., its repetend is 010989, so its period is 6, 6 divides into 91-1 evenly, and there is no smaller value of d with this period.
%e A350220 148 is a term since 1/148 = 0.00675675..., its repetend is 675, so its period is 3, 3 divides into 148-1 evenly, and there is no smaller value of d with this period.
%e A350220 Note that 370 is not in the sequence even though the repetend of 1/370 is 027 (period = 3) and 3 divides 370-1 because the period of 3 is accounted for by 148; note, 370 is in the related sequence A351396.
%o A350220 (Python)
%o A350220 from itertools import count, islice
%o A350220 from sympy import n_order, multiplicity, isprime
%o A350220 def A350220_gen(): # generator of terms
%o A350220     pset = set()
%o A350220     for d in count(1):
%o A350220         if not (isprime(d) or (p := n_order(10, d//2**multiplicity(2, d)//5**multiplicity(5, d))) <= 1 or (d-1) % p or p in pset):
%o A350220             yield d
%o A350220             pset.add(p)
%o A350220 A350220_list = list(islice(A350220_gen(),50)) # _Chai Wah Wu_, May 19 2022
%Y A350220 Cf. A007732 (digits period), A000010 (totient), A351396.
%K A350220 nonn,base
%O A350220 1,1
%A A350220 _Barry Smyth_, Mar 27 2022