A381946 a(n) is the smallest positive integer k with at least one digit > 1 such that k*n contains all the distinct digits of n.

12, 6, 12, 6, 3, 6, 21, 6, 21, 12, 12, 16, 24, 51, 7, 26, 42, 6, 48, 6, 6, 6, 14, 18, 5, 24, 26, 26, 32, 12, 23, 26, 4, 41, 9, 26, 19, 22, 24, 6, 4, 7, 8, 6, 9, 14, 31, 8, 6, 3, 3, 26, 25, 27, 3, 26, 65, 26, 5, 6, 24, 23, 22, 26, 21, 4, 25, 12, 14, 21, 17, 24, 19, 47, 5, 22, 14, 24, 25

Offset: 1

Author

M. F. Hasler and Ali Sada, Mar 10 2025

Keywords

• nonn
• base

Comments

We require that k has a digit > 1 in order to exclude "trivial" solutions like k = 1 or k = 10 or (if those are forbidden) k = 10^m + 1 where m is about half the length of n, e.g., k = 11 for most 2-digit numbers ab => 11*ab = a(a+b)b.

Examples

			26 is the smallest positive integer with digits greater than 1, and when multiplied by 16, it produces 416. The resulting product, 416, contains the digits 1 and 6, which are the distinct digits of 16. Therefore, a(16) = 26.
		

Crossrefs

Cf. A381700.

Programs

PARI

apply( {A381946(n)=my(S=Set(digits(n))); for(k=2, oo, #setminus(S, Set(digits(k*n))) || vecmax(digits(k))<2 || return(k))}, [1..99])

PARI

a(n) = my(d=digits(n), s=Set(d), k=2); while (!((#select(x->(x>1), digits(k)) >= 1) && (setintersect(Set(digits(k*n)), s) == s)), k++); k; \\ Michel Marcus, Mar 11 2025

Formula

a(n) = A381700(n)/n.

Extensions

More terms from Michel Marcus, Mar 11 2025