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

33, 55, 91, 99, 148, 165, 175, 246, 259, 275, 325, 370, 385, 451, 481, 495, 496, 505, 561, 592, 656, 657, 703, 715, 825, 909, 925, 1035, 1045, 1105, 1233, 1375, 1476, 1480, 1626, 1729, 1825, 1912, 2035, 2120, 2275, 2368, 2409, 2465, 2475, 2525, 2556, 2752, 2821

Offset: 1

Author

Barry Smyth, Mar 24 2022

Keywords

• nonn
• base

Comments

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.

Examples

			33 is a term since 1/33 = 0.030303..., its repetend is 03 so its period is 2, and 2 divides 33-1.
91 is a term since 1/91 = 0.010989010989..., its repetend is 010898 so its period is 6, and 6 divides 91-1.
925000 is a term since 1/925000 = 0.00000108108... has a repetend of 108 and a period of 3, and 3 divides 925000-1.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..662 from Barry Smyth)
• Barry Smyth, Are pseudoprimes hiding out among the composite reciprocals?

Crossrefs

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

Programs

Python

from itertools import count, islice
from sympy import n_order, multiplicity, isprime
def A351396_gen(startvalue=1): # generator of terms >= startvalue
    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)))
A351396_list = list(islice(A351396_gen(),50)) # Chai Wah Wu, May 19 2022