An interesting problem from Crux Mathematicorum, March 2004: The increasing sequence 1, 5, 6, 25, 26, 30, 31, 125, 126, … consists of positive integers that can be formed by adding distinct powers of 5. That is, 1 = 50, 5 = 51, 6 = 50 + 51, and so on. What’s the 75th integer in this sequence?
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Write the sequence in base 5:
1, 10, 11, 100, 101, 110, …
Now if we consider these expressions as a sequence of binary numbers, then the number in position n is just n — if we convert the sequence from binary to base 10 it’s just 1, 2, 3, 4 …
75 in binary is 1001011, so the number we want is 56 + 53 + 5 + 1 = 15,756.
(“Mathematical Mayhem,” Crux Mathematicorum 30:2 [March 2004], 72-76.)
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