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A347111
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a(n) is n in binary rendered as a base-10 number minus n in ternary rendered as a base-10 number.
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0
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0, 8, 1, 89, 89, 90, 90, 978, 901, 909, 909, 990, 990, 998, 991, 9879, 9879, 9810, 9810, 9898, 9891, 9899, 9899, 10780, 10780, 10788, 10011, 10099, 10099, 10100, 10100, 98988, 98981, 98989, 98989, 99000, 99000, 99008, 99001, 99889, 99889, 99890, 99890, 99978
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OFFSET
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1,2
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COMMENTS
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Theorem: a(6k - 2) = a(6k - 1), a(6k) = a(6k + 1), and a(6k + 2) != a(6k + 3) for all positive integers k.
Proof: a(6k - 2) in binary has a last digit of 0 and in ternary has a last digit of 1. a(6k - 1) is unchanged in both binary and ternary except for the last digit, which is increased by one in both bases. Since the 0 becomes a 1 in the binary number, which is a valid digit in binary, and the 1 becomes a 2 in the ternary number, which is a valid digit in ternary, the base-10 difference between the two numbers is the same.
The proof of the equality of a(6k) and a(6k + 1) is essentially the same; the binary and ternary numbers have last digits of 0, and those numbers plus 1 have last digits of 1; therefore the difference is the same.
In the last part of the theorem, a(6k + 2) != a(6k + 3), the binary number will increase by 1 when rendered in base 10. However, the ternary number's second to last digit will change because the last digit goes from 2 to 0. This means that there is an increase in the number not equal to one. Therefore the differences are different.
Theorem: a(n) contains only the digits 0, 1, 7, 8, and 9.
Proof: When we subtract the numbers in base 10, the possible digits of the difference are all of the possible combinations of the digits, including borrowed digits. With the number subtracted from being a number in binary and the number that is subtracted being a number in ternary, we can get the digits 0, 1, 7, 8, and 9 (with borrowing), and no other digits.
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LINKS
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FORMULA
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EXAMPLE
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a(7) = 111 (7 in binary) - 21 (7 in ternary) = 90.
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MATHEMATICA
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Array[Subtract @@ Map[FromDigits, IntegerDigits[#, {2, 3}]] &, 44] (* Michael De Vlieger, Aug 18 2021 *)
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PROG
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(PARI) a(n) = fromdigits(digits(n, 2)) - fromdigits(digits(n, 3)); \\ Michel Marcus, Aug 19 2021
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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STATUS
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approved
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