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A350598 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 repetend.

%I A350598 #43 Dec 25 2024 02:01:52
%S A350598 33,55,91,99,148,165,175,246,259,275,325,370,385,451,481,495,496,505,
%T A350598 561,592,656,657,703,715,825,909,925,1035,1045,1105,1233,1375,1476,
%U A350598 1626,1729,1825,1912,2035,2120,2275,2368,2409,2465,2475,2525,2556,2752,2821,2981,3160
%N A350598 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 repetend.
%C A350598 This is a subset of sequence A351396, and adds the extra condition that d is included only if it is the smallest value of d with a given repetend; thus duplicate repetends are not permitted. This eliminates some values of A351396 which are powers of 10 of d. For example, 1480 is excluded because although its period (k=3 based on a repetend of 675) divides evenly into 1479, this repetend already exists for a smaller value of d, namely 148, and 3 also divides evenly into 147. 1480 is the smallest such value of d from A351396 that will be excluded based on this modification. Other values of A351396 that are excluded include 3700, 5920, 9250, 14800, ...
%H A350598 Chai Wah Wu, <a href="/A350598/b350598.txt">Table of n, a(n) for n = 1..861</a>
%H A350598 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 A350598 33 is a term since 1/33 = 0.030303..., its repetend is 03, so its period is 2, 2 divides 33-1 evenly, and there is no smaller value of d with this repetend.
%e A350598 148 is in the sequence because 1/148 has 675 as its repetend, so its period is 3 and 3 divides 148-1.
%o A350598 (Python)
%o A350598 from itertools import count, islice
%o A350598 from sympy import n_order, multiplicity, isprime
%o A350598 def A350598_gen(): # generator of terms
%o A350598     pset = set()
%o A350598     for d in count(1):
%o A350598         if not isprime(d):
%o A350598             m2, m5 = multiplicity(2,d), multiplicity(5,d)
%o A350598             r = max(m2,m5)
%o A350598             k, m = 10**r, 10**(t := n_order(10,d//2**m2//5**m5))-1
%o A350598             c = k//d
%o A350598             s = str(m*k//d-c*m).zfill(t)
%o A350598             if not (t <= 1 or (d-1) % t or s in pset):
%o A350598                 yield d
%o A350598                 pset.add(s)
%o A350598 A350598_list = list(islice(A350598_gen(),50)) # _Chai Wah Wu_, May 19 2022
%Y A350598 Cf. A007732 (digits period), A000010 (totient), A351396.
%K A350598 nonn,base
%O A350598 1,1
%A A350598 _Barry Smyth_, Mar 27 2022