----- Original Message ----- From: "Eric Angelini" <keynews.tv@skynet.be> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Monday, July 04, 2005 7:35 AM Subject: [math-fun] Jump-my-digits Hello Math-fun fans, "Jump-my-digits" numbers. Take any integer of the sequence and repeat it as many times as you wish -- like this (for 258): 258,258,258,258,258,258,258,... Choose now any digit of 258, "2", for instance, and jump over the next 2 digits: you'll land on another "2". The same can be done with "5" and with "8": jumping respectively over 5 and 8 digits will see you land on another "5" or another "8". Question: What could be the smallest such number containing all 10 different digits? (0->9) If it doesn't exist, the smallest one containing 9 different digits, etc. Best, Ã. ------------------------------------------------------------- Let n be a JMD number. Let S be the repeated sequence of digits of n. Then if digit d occurs in n, then d occurs in S with frequency 1/(d+1). This means that SUM(d occurs in n, 1/(d+1)) <= 1. Since SUM(0 <= d <= 9, 1/(d+1)) > 1, this shows that JMD number n cannot include all 10 digits. Indeed, if a JMD number includes the digit 0, it cannot include any other digit. A second constraint on the digits that can occur in a JMD number is this: If distinct digits d and e occur in JMD number n, then gcd(d+1, e+1) > 1. This constraint means that many pairs of digits cannot occur in a JMD number, and limits the number of distinct digits in a JMD number to at most five, although there are further considerations that force even fewer distinct digits in a JMD number. Exhaustive analysis reveals that at most three distinct digits can occur in a JMD number. The possible sets of distinct digits that can occur in a JMD number are 0 1 13 137 15 17 19 2 25 258 28 3 35 37 39 4 49 5 57 59 6 7 79 8 9