Programming Praxis

Regular readers of Programming Praxis know of my affinity for prime numbers. Today’s exercise is based on a prime-generating sequence described by Eric Rowland. Consider the sequence a₁ = 7, a_n = a_n-1 + gcd(n, a_n-1): 7, 8, 9, 10, 15, 18, 19, 20, 21, 22, 33, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 69, 72, 73. Taking the differences between successive elements of the sequence gives a second sequence 1, 1, 1, 5, 3, 1, 1, 1, 1, 11, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 3, 1. All the elements of that sequence are either 1 or prime. Eliminating the 1s gives a third sequence of primes that begins 5, 3, 11, 3, 23, 3, 47, 3, 5, 3, 101, 3, 7, 11, 3, 13, 233, 3, 467, 3, 5, 3, 941, 3. Rowland has proved that all elements of the sequence are either 1 or prime; it is conjectured but not proven that all the odd primes appear in the sequence.

It is possible to shortcut the sequence by omitting the 1s, since the number of 1s at any point can be pre-computed. If we have a least-prime-divisor function lpd, then Vladimir Shevelev describes the sequence as a₁ = lpd(6-1) = 5, a₂ = lpd(6-2+5) = 3, a₃ = lpd(6-3+5+3) = 11, a₄ = lpd(6-4+5+3+11) = 3, a₅ = lpd(6-5+5+3+11+3) = 23, …, and a_n = lpd(6 – n + sum(a₁ … a_n-1).

Your task is to write functions that generate the three sequences, including the shortcut. 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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We have previously solved Part 1 and Part 2 of the Greplin Programming Challenge. In today’s exercise we will solve the third and final part:

Find all the subsets of a set of non-negative integers where the largest number is the sum of the remaining numbers, and return a count of the number of them. For instance, for the set { 1, 2, 3, 4, 6 } the subsets are 1+2=3, 1+3=4, 2+4=6, and 1+2+3=6, so the result is 4 subsets. Apply your program to the set { 3, 4, 9, 14, 15, 19, 28, 37, 47, 50, 54, 56, 59, 61, 70, 73, 78, 81, 92, 95, 97, 99 }.

Your task is to write a program to solve the challenge. In addition to solving the challenge, you might like to apply your solution to the set of prime numbers less than 2¹⁰. 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 internet provides a rich source of data for those who are interested: stock prices, baseball statistics, the littoral area in acres of lakes in Minnesota, and much more. In the United States, the National Oceanic and Atmospheric Administration publishes daily weather forecasts at http://weather.noaa.gov/pub/data/forecasts/.

Your task is to write a program that retrieves the current weather forecast for a requested city. 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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An emirp is a prime number that is also prime when its digits are reversed, and that is not also a palindrome. For instance, 13 is an emirp because its reversal, 31, is also prime; 23 is not an emirp, even though it is prime, because its reversal, 32, is not prime; and 101 is not an emirp, even though it is prime, because it is a palindrome.

Your task is to enumerate the emirps below a million; you should strive for maximum speed. 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 recent programming challenge asked you to solve the following problem:

Find the smallest prime fibonacci number greater than a given input number. Add one to that fibonacci prime, find the factors of the result, and return the sum of the factors. For instance, if the input number is 10, the smallest fibonacci prime greater than 10 is 13, the factors of 13 + 1 = 14 are 2 and 7, and their sum is 9.

Your task is to write a function to solve that problem, then apply your function to the input number 227000. 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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Benford’s Law, which was discovered by Simon Newcomb in 1881 and rediscovered by Frank Benford in 1938, states that, for many sets of numbers that arise in a scale-invariant manner, the first digit is distributed logarithmically, with the first digit 1 about 30% of the time, decreasing digit by digit until the first digit is 9 about 5% of the time. Stated mathematically, the leading digit d ∈ {1 … b-1} for a number written in base b will occur with probability P_d = log_b(1 + ¹/_d). Thus, in base 10, the probabilities of the first digit being the number 1, 2, … 9 are 30.1%, 17.6%, 12.5%, 9.7%, 7.9%, 6.7%, 5.8%, 5.1% and 4.6%.

Benford’s Law is counter-intuitive but arises frequently in nature. It is also frequently used in auditing. Make a list of the amounts of the checks that a bookkeeper has written in the past year; if more that 5% begin with the digit 8 or 9, the bookkeeper is likely an embezzler. More important, precinct results of the disputed Iranian elections a year ago displayed anomalous first-digit counts, suggesting vote fraud.

Recently, Shriram Krishnamurthy issued a programming challenge on the Racket mailing list, asking for smallest/tightest/cleanest/best code to calculate the first-digit percentages of a list of numbers; he also challenged readers to apply the function to data in a comma-separated values file. He didn’t give a source, but did mention that he was interested in the littoral area (in acres) of the lakes of Missesota; sample data appears on the next page.

Your first task is to write a function that calculates the first-digit percentages of a list of numbers. Your second task is to calculate the first-digit percentages of the data on the next page. 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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In the previous exercise we developed four functions for reading records from various type of text-file databases. In today’s exercise we develop four functions for processing those records, including map, fold, filter, and for-each. We also wrote map-reduce in a previous exercise, which gives us a fifth processing function.

All four functions take as their first argument a reader function that fetches the next record from the input; we wrote some reader functions in the previous exercise, and it is common to write others for specific input formats. Map takes both a reader function and a transformer function and applies the transformer to each input record, returning a list of the transformed values in the same order as the input. Fold takes a reader function, a combining function and a base value and applies the combiner successively to each base value and the next record from the input, returning the final base value when the input is exhausted. Filter is a combinator; it takes a reader function and a predicate and returns a new reader function that passes only those input records for which the predicate is true. For-each takes a reader function and another procedure and applies the procedure to each input record in turn, only for its side-effects, until the input is exhausted; it returns nothing.

Your task is to write the four functions for processing text-file database records. 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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There is a lot of data stored in plain-ascii text files consisting of records separated by newlines, each record consisting of multiple fields, and it is useful to have a function library for dealing with them. This exercise looks at some functions for reading the data; the next exercise will look at some functions for processing the data.

We will consider four common types of text file databases. A file with fixed-length data fields has records of a fixed number of characters, each record containing fields that are similarly in fixed positions; the data may be preceded by a fixed-length header. A file with character-delimited fields has variable-length records, each with fields separated by a single-character delimiter; the delimiter is often a tab or vertical bar. A particular type of variable-length delimited text database is known as comma-separated values, where the delimiter is a comma and fields may be surrounded by double-quote characters so that a comma within a quoted field loses its meaning as a field separator; in that case, a literal double-quote character may appear within a quoted field as two double-quote characters in succession. The fourth type that we will consider is a name-value record, where each record consists of multiple fields, one field per line, separated by blank lines, each field consisting of a type-name and a value separated by a delimiter; this format is often used for databases that have many optional fields, such as bibliographic databases.

We want reader functions for each of these file formats that all return a single record each time they are called, or an end-of-file marker when the input is exhausted, and advance the file pointer to the beginning of the next record. The return value should be a list or array, whichever is convenient, containing the value of one field in each element, except for the name-value record, which should return a list of name/value pairs.

Different operating systems have different methods of signalling the end of a line. For maximum portability, your functions should accept the end of a line indicated by a carriage return, a line feed, or both characters in either order. You should be prepared to accept any type of line marker because the data may come from any source; for instance, your computer running MS Windows with a CRLF line marker may fetch data from a Linux computer with a bare LF for the line marker. You should also accept the final line in the file whether or not it has a trailing line marker.

Your task is to write functions to read one record from each of the four file types described above. 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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Greplin issued a programming challenge recently that required programmers to solve three problems; when completed, Greplin issued an invitation to send them a resume. The first problem required the programmer to find the longest palindrome in the following 1169-character string:


Fourscoreandsevenyearsagoourfaathersbroughtforthonthisconta
inentanewnationconceivedinzLibertyanddedicatedtotheproposit
ionthatallmenarecreatedequalNowweareengagedinagreahtcivilwa
rtestingwhetherthatnaptionoranynartionsoconceivedandsodedic
atedcanlongendureWeareqmetonagreatbattlefiemldoftzhatwarWeh
avecometodedicpateaportionofthatfieldasafinalrestingplacefo
rthosewhoheregavetheirlivesthatthatnationmightliveItisaltog
etherfangandproperthatweshoulddothisButinalargersensewecann
otdedicatewecannotconsecratewecannothallowthisgroundThebrav
elmenlivinganddeadwhostruggledherehaveconsecrateditfarabove
ourpoorponwertoaddordetractTgheworldadswfilllittlenotlenorl
ongrememberwhatwesayherebutitcanneverforgetwhattheydidhereI
tisforusthelivingrathertobededicatedheretotheulnfinishedwor
kwhichtheywhofoughtherehavethusfarsonoblyadvancedItisrather
forustobeherededicatedtothegreattdafskremainingbeforeusthat
fromthesehonoreddeadwetakeincreaseddevotiontothatcauseforwh
ichtheygavethelastpfullmeasureofdevotionthatweherehighlyres
olvethatthesedeadshallnothavediedinvainthatthisnationunsder
Godshallhaveanewbirthoffreedomandthatgovernmentofthepeopleb
ythepeopleforthepeopleshallnotperishfromtheearth


Your task is to write a function that finds the longest palindrome in a string and apply it to the string given above. 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 exercise is simple but tricky: write a function to rotate the elements of an array. The function takes two arguments: the array to be rotated and the number of elements to rotate, where a positive count rotates to the left and a negative count rotates to the right. For instance, given the array [1 2 3 4 5 6], a rotation of 2 to the left gives [3 4 5 6 1 2], and a further rotation of 2 to the right restores the original array.

You should be sure to handle the edge cases properly. If the count is zero or the length of the array, the array should remain unchanged. If the count is greater than the length of the array, you should still do the right thing; for instance, a rotation of 8 on the array given above gives [3 4 5 6 1 2], the same as a rotation of 2.

Your task is to write the rotate function described above. 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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