Programming Praxis

[ Today’s exercise will be the last until next year. Expect the next exercise the week of January 6. In the meantime, enjoy the Christmas and New Year’s holidays with your families. ]

A set is an unordered collection of elements, and supports two operations: add an element, and delete an element. A set without replacement is a set where an element, once deleted from the set, cannot later be added back into the set.

Your task is to implement sets without replacement. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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The sequence of pentabonacci numbers is defined as the sequence beginning 0, 1, 1, 2, 4, and continuing with each subsequent number the sum of the five previous members of the sequence. The sequence grows very quickly; the 20th member of the sequence (counting the 0 that begins the sequence as the first member) is 103519.

Your task is to write a program that calculates the first n members of the pentabonacci sequence and determine the rule that states whether the nth member of the sequence is even or odd. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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Regular readers know of my affinity for the Online Encyclopedia of Integer Sequences. I was browsing the other day and came across A052155, the sequence of pseudoprimes to both base2 and base3. This sequence is important because it forms the basis of a very fast primality test for numbers in the range 2³² to 2⁶⁴, which is a very important range for some large factoring algorithms (particularly SIQS and NFS in their 2LP variants): a number n in range is prime if 2^n-1 ≡ 1 (mod n), 3^n-1 ≡ 1 (mod n), and n is not in a small list of known pseudoprimes to bases 2 and 3. The two modular exponentiations are reasonably quick, the lookup by binary search is even quicker, and the primality test is fully deterministic.

Your task is to calculate the pseudoprimes to bases 2 and 3 less than a million. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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A radix-10 number is a Muenchhausen number if it is equal to the sum of its digits, each digit raised to the power of itself. For instance, 3435 is a Muenchhausen number because 3**3 + 4**4 + 3**3 + 5**5 = 3435. Strict Muenchhausen numbers do not permit 0 as a digit, because 0**0 is undefined. Variant Muenchhausen numbers permit 0 as a digit, with 0**0 defined as 0. Muenchhausen numbers also exist with radices other than 10.

Your task is to find all the Muenchhausen numbers in radix-10. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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Today’s task challenges you write a useful program using only a single line of code. It is not fair to abuse the notion of a line. You can probably do more than you expect.

Your task is to write a useful program using only a single line of code. When you are finished, you are welcome to read a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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