Divisors

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

General characteristics

Domain and analyticity

divisors(n) is a nonanalytical function which is defined over ℤ.

Symmetries and periodicities

Parity

divisors(n) is an even function.

Periodicity

No periodicity

Products

Finite products

Operations

Representations through equivalent functions

With related functions

Theorems

The squares problem

Special cases of the s squares problem to find the number (Subscript[r, s])(n) of integer solutions of (Subscript[x, 1]^2)+(Subscript[x, 2]^2)+…+(Subscript[x, s]^2)==n (order and sign changes are taken into account) can be expressed through the divisors of n:

(Subscript[r, 4])(n)=8(∑_(Underscript[d|n, d/4 ∉ℕ]))d;  
(Subscript[r, 8])(n)=16(∑_(d|n))(((-1))^(n+d))(d^3);
(Subscript[r, 16])(n)=-(((-1))^n)(32/3)((Subscript[(OverTilde[σ]), 1])(n)+(Subscript[(OverTilde[σ]), 3])(n)+(Subscript[(OverTilde[σ]), 5])(n))+(((-1))^n)(256/3)(∑_(k=1)^(n-1))((Subscript[(OverTilde[σ]), 1])(k)(Subscript[(OverTilde[σ]), 5])(n-k)-(Subscript[(OverTilde[σ]), 3])(k)(Subscript[(OverTilde[σ]), 3])(n-k))/;(Subscript[(OverTilde[σ]), r])(n)==(∑_(d|n))(((-1))^(d+n/d))(d^r);

(Subscript[r, 24])(n)=(((-1))^n)(16/9)(17(Subscript[(OverHat[σ]), 3])(n)+8(Subscript[(OverHat[σ]), 5])(n)+2(Subscript[(OverHat[σ]), 7])(n))+(((-1))^n)(512/9)(∑_(k=1)^(n-1))((Subscript[(OverHat[σ]), 3])(k)(Subscript[(OverHat[σ]), 7])(n-k)-(Subscript[(OverHat[σ]), 5])(k)(Subscript[(OverHat[σ]), 5])(n-k))/;(Subscript[(OverHat[σ]), r])(n)==(∑_(d|n))(((-1))^d)(d^r)

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