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.

33, 91, 148, 246, 451, 496, 505, 561, 657, 703, 1035, 1105, 1912, 2120, 2465, 2556, 2752, 2821, 4005, 4141, 5461, 6525, 6533, 6565, 6601, 6700, 7107, 8695, 8905, 8911, 10585, 11649, 12403, 12801, 13366, 13695, 13833, 14701, 15211, 15841, 17120, 18336, 19345, 19503, 19900

Offset: 1

Author

Barry Smyth, Mar 27 2022

Keywords

• nonn
• base

Comments

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).

Examples

			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.
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.
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.
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.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..3839
• Barry Smyth, Are pseudoprimes hiding out among the composite reciprocals?, Towards Data Science, Mar 25 2022.

Crossrefs

Cf. A007732 (digits period), A000010 (totient), A351396.

Programs

Python

from itertools import count, islice
from sympy import n_order, multiplicity, isprime
def A350220_gen(): # generator of terms
    pset = set()
    for d in count(1):
        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):
            yield d
            pset.add(p)
A350220_list = list(islice(A350220_gen(),50)) # Chai Wah Wu, May 19 2022