A381527 a(n) is the smallest positive multiple of n such that the decimal expansion of a(n)/n is a substring of the decimal expansion of a(n), or -1 if no such number exists.

-1, 1, -1, 15, -1, 25, 12, 35, -1, 45, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, -1, 21, 198, 115, 1728, 125, 52, 135, 168, 145, 150, 31, 2912, 165, 1428, 140, 72, 148, 1368, 195, -1, 41, 3948, 215, 132, 135, 92, 235, 384, 245, 250, 51, 1248, 265, 378, 275

Offset: 0

Author

Dominic McCarty, Feb 26 2025

Keywords

• sign
• easy

Examples

			44*1 = 44 does not have 1 as a substring.
44*2 = 88 does not have 2 as a substring.
44*3 = 132 has 3 as a substring.
So, a(44) = 132.
		

Links

• Dominic McCarty, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A381526 (for a(n)/n).

Programs

Python

from itertools import count
def a(n):
    if n in [0, 2, 4, 8]: return -1
    if n % 10 == 0: return -1 if (k:=a(n//10)) == -1 else 10*k
    for k in count(1):
        if str(k) in str(k*n): return k*n
print([a(n) for n in range(100)])

Formula

a(10*n) = 10*a(n) for a(n) != -1

a(n) = A381526(n)*n for A381526(n) != -1