Somewhat related to the curves shown here is the function MoebiusMu[n] , equal to 0 if n has a repeated prime factor and otherwise (-1)^Length[FactorInteger[n]] .

The picture on page 613 shows the values of m and n for which this is equal to n .

The sequence {1, 2, 2, 1, 1, 2, …} defined by the property list  Map[Length, Split[list]] was suggested as a mathematical puzzle by William Kolakoski in 1965 and is equivalent to Join[{1, 2}, Map[First, CTEvolveList[{{1}, {2}}, {2}, t]]] It is known that this sequence does not repeat, contains no more than two identical consecutive blocks, and has at least very close to equal numbers of 1's and 2's.

The operator can be either Xor or Equal (6 or 9).

New expressions are also created by replacing each possible variable with x ⊼ y , where x and y are new variables, and by setting every possible pair of variables equal in turn.

First, as equations which can be manipulated—like the axioms of basic logic—to establish whether expressions are equal.

Leading digits [in numbers] Even though in individual numbers generated by simple mathematical procedures all possible digits often appear to occur with equal frequency, leading digits in sequences of numbers typically do not.

The evolution of the system for t steps can be obtained from SSEvolve[rule_, init_, t_, d_Integer] := Nest[FlattenArray[# /. rule, d] &, init, t] FlattenArray[list_, d_] := Fold[Function[{a, n}, Map[MapThread[Join, #, n] &, a, -{d + 2}]], list, Reverse[Range[d] - 1]] The analog in 3D of the 2D rule on page 187 is {1  Array[If[LessEqual[##], 0, 1] &, {2, 2, 2}], 0  Array[0 &, {2, 2, 2}]} Note that in d dimensions, each black cell must be replaced by at least d + 1 black cells at each step in order to obtain an object that is not restricted to a dimension d - 1 hyperplane.

Properties [of number theoretic sequences] (a) The number of divisors of n is given by DivisorSigma[0, n] , equal to Length[Divisors[n]] . … (b) (Aliquot sums) The quantity that is plotted is DivisorSigma[1, n] - 2n , equal to Apply[Plus, Divisors[n]] - 2n . … The total number of ways that integers less than n can be expressed as a sum of d squares is equal to the number of integer lattice points that lie inside a sphere of radius Sqrt[n] in d -dimensional space.

One starts by setting r equal to p .