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A108641
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"Jump-my-digits" numbers.
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1
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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
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refs;
listen;
history;
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internal format)
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OFFSET
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1,2
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COMMENTS
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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". What could be the smallest such number containing 10 different digits? If it doesn't exist, the smallest containing 9 different digits, etc.
No "jump-my-digits" number in decimal can have more than 3 distinct digits. Look at the digits plus 1, this being the frequency at which the digit appears in the iterated number. Obviously, no two such incremented digits can be relatively prime. Only choosing 2 as the GCD provides more than 3 possible digits. By the same argument, of 2,4,6,8,10, only 4 and 8 can occur in positions congruent modulo 2, so at most 3 distinct digits are possible. - Franklin T. Adams-Watters, Sep 27 2006
The sequence is trivially infinite, as the concatenation of any term with itself is a term. - Franklin T. Adams-Watters, Sep 27 2006
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LINKS
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CROSSREFS
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KEYWORD
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base,easy,nonn
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AUTHOR
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STATUS
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approved
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