A360423 Positive integers n (with k digits) such that if a positive integer m with k+1 digits is divisible by n, then all the rotations of m are divisible by n.

1, 3, 9, 27, 37, 101, 303, 909, 2439, 10101, 10989, 12987, 15873, 25641, 27027, 30303, 37037, 47619, 76923, 90909, 1010101, 1369863, 3030303, 9090909, 12345679, 27027027, 37037037, 101010101, 243902439, 303030303, 909090909, 10101010101, 10989010989, 12987012987, 15873015873

Offset: 1

Author

Robert C. Lyons, Feb 14 2023

Keywords

• nonn
• base

Comments

John D. Cook's blog (see link below) provides a proof that "if a three-digit number is divisible by 37, it remains divisible by 37 if you rotate its digits."

Examples

			For a(4)=27, 405 is a 3-digit multiple of 27, and the two rotations of 405 (i.e., 54 and 540) are also multiples of 27.
For a(5)=37, 185 is a 3-digit multiple of 37, and the two rotations of 185 (i.e., 851 and 518) are also multiples of 37.
For a(9)=2439, 12195 is a 5-digit multiple of 2439, and the four rotations of 12195 (i.e., 21951, 19512, 95121 and 51219) are also multiples of 2439.
		

Links

• John D. Cook, Rotating multiples of 37.

Crossrefs

Cf. A034089, A066484.

Programs

Python

def rotate(str):
    first_char = str[0 : 1]
    remaining_chars = str[1 :]
    return (remaining_chars + first_char)
def get_rotations(n):
    n_as_str = str(n)
    rotations = []
    rotation_as_str = n_as_str
    for i in range(len(n_as_str) - 1):
        rotation_as_str = rotate(rotation_as_str)
        rotations.append(int(rotation_as_str))
    return rotations
seq = []
max_n = 9999999
for n in range(1, max_n + 1):
    n_len = len(str(n))
    factor = 2
    while True:
        prod = n * factor
        prod_len = len(str(prod))
        if prod_len < n_len + 1:
            factor = factor + 1
        elif prod_len > n_len + 1:
            seq.append(n)
            break
        else:
            # prod_len == n_len + 1
            rotations = get_rotations(prod)
            if all(rotation % n == 0 for rotation in rotations):
                factor = factor + 1
            else:
                break
print(seq)

Extensions

a(25)-a(35) from Chai Wah Wu, Feb 24 2023