[math-fun] Jump-my-digits
Hello Math-fun fans, I've just submitted this to Sloane's OEIS: 1 2 3 4 5 6 7 8 9 11 13 15 17 19 22 31 33 35 37 39 44 51 53 55 57 59 66 71 73 75 77 79 88 91 93 95 97 99 111 222 225 228 252 255 258 282 285 288 333 444 522 525 528 552 555 558 582 585 666 777 822 825 828 852 855 882 888 999 1111 1313 1317... "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, É.
Eric Angelini wrote:
Hello Math-fun fans,
I've just submitted this to Sloane's OEIS:
1 2 3 4 5 6 7 8 9 11 13 15 17 19 22 31 33 35 37
What about 0? <snip>
"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".
If a_{0 mod 3} = 2, a_{1 mod 3}, and a_{2 mod 3}=8, this doesn't amount to a_{i+a_i}=a_i, which is what I expected on the first reading. Your definition gives a_{i+a_i+1}=a_i. If you use the definition I expected, then 20 can be included, since there's a period 2 signal and 0 goes back to itself. For that sequence, I get 0,1,2,3,4,5,6,7,8,9,11,20,22,24,26,28,33,...
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, É.
Well, with your definition, 0 can't show up except in 0. In mine, 1 can't show up except in repdigits of 1. In mine, 1<gcd(d,d') for all pairs of digits d,d' (where gcd(x,0)=oo) and their indices have to take different values modulo their gcd. Also, 1<gcd(d,period) unless period=1. That means 02428242 is my winner. In yours, 1<gcd(d+1,d'+1) and 1<gcd(d+1,period) unless period=1. So you can have at most three different digits. That means 258 is your winner. Unless I've made a mistake, you can't get a number with four different digits. (Unless, of course, you switch to a different base.) -- Mike Stay staym@clear.net.nz http://www.cs.auckland.ac.nz/~msta039
----- 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
participants (3)
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David Wilson -
Eric Angelini -
M. Stay