A382137 Smallest integer that cannot be converted to a multiple of n by changing at most one of its decimal digit.

545, 51, 44, 31, 21, 21, 22, 21, 21, 11, 15, 11, 11, 11, 11, 11, 12, 11, 11, 11, 14, 11, 11, 11, 11, 11, 11, 11, 11, 11, 13, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12

Offset: 11

Author

Mickaël Launay, Mar 27 2025

Keywords

• nonn
• base

Comments

a(n) is not defined for n<11 since any integer can be converted to a multiple of n by changing its last digit.

For all n, a(n)>=11, since any number <=10 ca be converted to 0 which is a multiple of n.

For n>=42, a(n)=11, except for a(51)=a(61)=a(71)=a(81)=a(91)=12. Especially a(n)=11 for all n>=100 because 11 is the first integer that cannot be converted to 0 or a 3-digit number.

Examples

			For n=12, any number<50 can be converted to a multiple of 12 by changing its last digit. Since a) there is no multiple of 12 between 50 and 59 so changing the last digit is useless, b) 50 can be converted to 60=5*12 and c) changing the "5" of 51 gives only odd (non multiples of 12) numbers, then a(12)=51.
		

Links

• Mickaël Launay, Le démon des multiples, L’énigme maths du "Monde" n°48. In French.

Crossrefs

Cf. A353023, A192545.

Programs

Python

import math
def test(N,M):
   t = True
   for i in range(0,math.floor(math.log(N,10))+1):
      for j in range(0,10): t = t and ((N+(j-N // 10**i % 10)*10**i)%M != 0)
   return t
for M in range(11,100):
   N = 1
   while (not test(N,M)): N += 1
   print(N, end=', ')