A365933 a(n) is the period of the remainders when repdigits are divided by n.

1, 9, 27, 9, 9, 27, 54, 9, 81, 9, 18, 27, 54, 54, 27, 9, 144, 81, 162, 9, 54, 18, 198, 27, 9, 54, 243, 54, 252, 27, 135, 9, 54, 144, 54, 81, 27, 162, 54, 9, 45, 54, 189, 18, 81, 198, 414, 27, 378, 9, 432, 54, 117, 243, 18, 54, 162, 252, 522, 27, 540, 135, 162, 9, 54

Offset

1,2

Comments

For n>1: Periods are divisible by 9 (= a full cycle in the sequence of repdigits). a(n)/9 is the period of the remainders when repunits are divided by n. So the digit part of the repdigits has no effect on periods generally. For most n the beginning of the periodic part is always A010785(1). If n is a term of A083118 the periodic part starts later after some initial remainders that do not repeat.

Links

Table of n, a(n) for n=1..65.

Karl-Heinz Hofmann, Table with additional information.

Example

For n = 6:                Remainders of A010785(1..54) mod n.
A010785( 1...9) mod n:      [1, 2, 3, 4, 5, 0, 1, 2, 3]
A010785(10..18) mod n:      [5, 4, 3, 2, 1, 0, 5, 4, 3]
A010785(19..27) mod n:      [3, 0, 3, 0, 3, 0, 3, 0, 3]
So the period is 3*9 = 27. Thus a(n) = 27. And the pattern seen above starts again:
A010785(28..36) mod n:      [1, 2, 3, 4, 5, 0, 1, 2, 3]
A010785(37..45) mod n:      [5, 4, 3, 2, 1, 0, 5, 4, 3]
A010785(46..54) mod n:      [3, 0, 3, 0, 3, 0, 3, 0, 3]

Prog

(Python)
def A365933(n):
    if n == 1: return 1
    remainders, exponent = [], 1
    while (rem:=(10**exponent // 9 % n)) not in remainders:
        remainders.append(rem); exponent += 1
    return (exponent - remainders.index(rem) - 1) * 9
(Python)
def A365933(n):
    if n==1: return 1
    a, b, x, y=1, 1, 1%n, 11%n
    while x!=y:
        if a==b:
            a<<=1
            x, b=y, 0
        y = (10*y+1)%n
        b+=1
    return 9*b # Chai Wah Wu, Jan 23 2024

Crossrefs

Cf. A010785, A002275.

Cf. A305322 (divisor 3), A002279 (divisor 5), A366596 (divisor 7).

Cf. A083118 (the impossible divisors).

Sequence in context: A103952 A103955 A359531 * A109041 A227900 A010817

Adjacent sequences: A365930 A365931 A365932 * A365934 A365935 A365936

Keyword

nonn,base

Author

Karl-Heinz Hofmann, Nov 07 2023

Status

approved