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

1,1

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.

Links

Table of n, a(n) for n=1..79.

Formula

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

Example

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.

Prog

(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

Crossrefs

Cf. A381700.

Sequence in context: A283880 A084067 A240537 * A227354 A328043 A075247

Adjacent sequences: A381943 A381944 A381945 * A381947 A381948 A381949

Keyword

nonn,base

Author

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

Extensions

More terms from Michel Marcus, Mar 11 2025

Status

approved