Divisors The picture below shows as black squares the divisors of each successive number (which correspond to the gray dots in the picture in the main text). Primes have divisors 1 and n only.

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 number of ways of expressing an integer n as the sum of two such squares is 4 Apply[Plus, Im[  ^Divisors[n]]] .

Iterated aliquot sums Related to case (b) above is a system which repeats the replacement n  Apply[Plus, Divisors[n]] - n or equivalently n  DivisorSigma[1, n] - n .

With any such shift rule, all states lie on cycles, and the lengths of these cycles are the divisors of the size n . Every cycle corresponds in effect to a distinct necklace with n beads; with k colors the total number of these is Apply[Plus, (EulerPhi[n/#] k # &)[Divisors[n]]]/n The number of cycles of length exactly m is s[m, k]/m , where s[m, k] is defined on page 950 .

Perfect numbers Perfect numbers with the property that Apply[Plus, Divisors[n]]  2n have been studied since at least the time of Pythagoras around 500 BC. … Various generalizations of perfect numbers have been considered, requiring for example IntegerQ[DivisorSigma[1, n]/n] (pluperfect) or Abs[DivisorSigma[1, n] - 2n] < r (quasiperfect).

Repetition in numbers A common source of repetition in systems involving numbers is the almost trivial fact that in a sequence of successive integers there is a repetitive pattern of cases at which a particular divisor occurs.

The number of states with spatial period m is given by s[m_, k_]:= k m - Apply[Plus, Map[s[#, k] &, Drop[Divisors[m], -1]]] or equivalently s[m_, k_]:=Apply[Plus, (MoebiusMu[m/#] k # &)[Divisors[m]]] In a cellular automaton with a total of n cells, the maximum possible repetition period is thus s[n, k] .

Typically the reason is that the states in each piece are shifted copies of each other, and in such cases the number of pieces will be a divisor of n .

(In 348 BC, Plato mentioned divisors of 5040, and by 100 AD there is evidence that the fifth perfect number was known, requiring the knowledge that 8191 is prime.)

(e) The pattern essentially shows which x are divisors of y , just as on pages 132 and 909 .