A378865 a(n) is the smallest positive integer k such that n*k uses none of the digits of n, or 0 if no such k exists.

2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 2, 3, 2, 2, 2, 2, 2, 2, 2, 0, 3, 2, 2, 4, 4, 3, 2, 2, 2, 0, 2, 2, 2, 2, 2, 2, 3, 2, 2, 0, 2, 4, 2, 2, 2, 2, 4, 2, 12, 0, 4, 2, 2, 2, 2, 2, 2, 2, 2, 0, 4, 5, 3, 2, 2, 2, 2, 3, 2, 0, 4, 2, 2, 3, 4, 2, 2, 2, 2, 0, 3, 2, 2, 3

Offset: 1

Author

Gonzalo Martínez, Dec 09 2024

Keywords

• nonn
• base

Comments

It is observed that a(10*m) = 0 and a(100*m + 5) = 0 for all positive integers m.

If m is a repdigit number (A010785) that does not have the digit 9, then a(m) = 2 and if m = 99...9, with t 9's, then a(m) = 11...2, i.e., (t - 1) 1's followed by 2, since 99...9 * 11...2 equals (t - 1) 1's followed by (t - 1) 8's, where k = 11...2 is the smallest number with this property. In other words, a(A002283(m)) = A047855(m), for all positive integers m.

Examples

			a(12) = 3, since 12*1 = 12, 12*2 = 24 have digits in common with 12, while 12*3 = 36 does not.
		

Links

• Dominic McCarty, Table of n, a(n) for n = 1..1000 (synthesized from _Michel Marcus_'s data at A074157)

Crossrefs

Cf. A074157, A010785, A002283, A047855.

Formula

a(n) = A074157(n)/n.