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Wolfram Mathematica

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Wolfram Mathematica, from Wolfram Research Inc., is a computer algebra system used for numerical and symbolic computations. It also has word processing and typesetting features, as the program is intended to be not just a computational engine but also a tool for documenting mathematical research—hence Mathematica documents are called "notebooks" and have a *.nb file extension. (Starting with Mathematica 8, one can also save notebooks as "Computable Documents" that have a *.cdf (Computable Document Format) file extension and leverage "the power and flexibility of the Mathematica language with the wide distribution provided by a public format.") Dynamic syntax highlighting was introduced in version 7.0. In this example, we see gray for comments, blue for symbols not yet defined, black for defined symbols, matched brackets and numeric literals, green for local symbols, and purple for unmatched brackets.

Some number theoretic functions

Prime numbers

The basic prime-related functions in Mathematica are Prime[n], PrimePi[x] and PrimeQ[n], which are the $n\,$ th prime function, the prime counting function $\pi (x)\,$ and the characteristic function of prime numbers respectively.

 PrimePi[998.7654] 168 Prime 997 PrimeQ[%] True

In that last example we showed %, the shortcut for the previous output, like the ANS key on a calculator. Closely allied to these functions is FactorInteger[n], which returns a list of prime factors with exponents, and, if necessary, a complex unit.

( ===== THIS SPACE RESERVED FOR SOME STUFF ABOUT PARTITION FUNCTIONS ===== )
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Some sequence handling functions

In the Mathematica documentation, sequence handling is referred to as "list manipulation."

The workhorse of sequence handling is perhaps the Table command, which takes a function, usually of one or two variables, and iterates it through some specified values for the variables.

This example gives a few small positive integers of the form $6k+1\,$ (the default starting value in Mathematica is 1; to start with 0 it must be explicitly stated)

 Table[6k + 1, {k, 20}] {7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103, 109, 115, 121}

Another list manipulation command frequently used in the OEIS is Select, which goes through a list and picks out those elements that meet the specified criterion. After the previous Table output, we could do

 Select[%, PrimeQ] {7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109}

Select is often paired with the humble and unassuming Range command. At first it would seem all Range does is give a list of consecutive integers.

 Range {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

But the starting value doesn't have to be 1, nor does 1 have to be the increment value, which can even be negative if desired (in which case it's a decrement). Thus,

 Range[100, 2, -7] {100, 93, 86, 79, 72, 65, 58, 51, 44, 37, 30, 23, 16, 9, 2}

gives A115020. But more usefully still, arithmetic operations can be applied to the entire sequence specifying each only once. Thus, our earlier example of $6k+1\,$ , can be more compactly given as 6Range + 1.

There are also commands for comparing sequences, finding the elements in common, etc.

EISFormat.m

EISFormat.m formats integer sequences in the native format expected by Neil Sloane's online encyclopedia of integer sequences. Written by Olivier Gerard and Eric W. Weisstein.