cp's OEIS Frontend

A351396 Composite numbers d such that the period k of the decimal expansion of 1/d is > 1 and divides d-1.

%I A351396 #172 May 20 2022 21:34:08
%S A351396 33,55,91,99,148,165,175,246,259,275,325,370,385,451,481,495,496,505,
%T A351396 561,592,656,657,703,715,825,909,925,1035,1045,1105,1233,1375,1476,
%U A351396 1480,1626,1729,1825,1912,2035,2120,2275,2368,2409,2465,2475,2525,2556,2752,2821
%N A351396 Composite numbers d such that the period k of the decimal expansion of 1/d is > 1 and divides d-1.
%C A351396 For primes p, the period k of the decimal expansion of 1/p divides p-1. This is usually not the case for reciprocals of composites d; instead, the period k always divides phi(d) where phi is Euler's totient function (A000010). This sequence lists the composites d for which k also divides d-1, which satisfies the condition of a pseudoprime, making such composites a sequence of pseudoprimes with respect to the divisibility of d-1 by k.
%H A351396 Chai Wah Wu, <a href="/A351396/b351396.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..662 from Barry Smyth)
%H A351396 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>
%e A351396 33 is a term since 1/33 = 0.030303..., its repetend is 03 so its period is 2, and 2 divides 33-1.
%e A351396 91 is a term since 1/91 = 0.010989010989..., its repetend is 010898 so its period is 6, and 6 divides 91-1.
%e A351396 925000 is a term since 1/925000 = 0.00000108108... has a repetend of 108 and a period of 3, and 3 divides 925000-1.
%o A351396 (Python)
%o A351396 from itertools import count, islice
%o A351396 from sympy import n_order, multiplicity, isprime
%o A351396 def A351396_gen(startvalue=1): # generator of terms >= startvalue
%o A351396     return filter(lambda d: not (isprime(d) or (p := n_order(10, d//2**multiplicity(2, d)//5**multiplicity(5, d))) <= 1 or (d-1) % p), count(max(startvalue,1)))
%o A351396 A351396_list = list(islice(A351396_gen(),50)) # _Chai Wah Wu_, May 19 2022
%Y A351396 Cf. A007732 (digits period), A000010 (totient).
%K A351396 nonn,base
%O A351396 1,1
%A A351396 _Barry Smyth_, Mar 24 2022