The Carmichael function lambda(n); its value equals the exponent of znstar.

The Carmichael function lambda(n), returned as a factorization sequence.

The divisor function sigma_i(n), which equals the sum of all the d^i for d dividing n, for integer n and small non-negative integer i.

The number of divisors of the positive integer n. This is a special case of DivisorSigma.

The sum of the divisors of the positive integer n. This is a special case of DivisorSigma.

The Euler totient function phi(n); its value equals the order of znstar.

EulerPhi(n) : RngIntElt -> RngIntElt

The Euler totient function phi(n); its value equals the order of znstar.

The Euler totient function phi(n), returned as a factorization sequence.

The Legendre symbol ((n/m)): for prime m this checks whether or not n is a quadratic residue modulo m. The function returns 0 if m divides n, -1 if n is not a quadratic residue, and 1 if n is a quadratic residue modulo m. A fast probabilistic primality test is performed on m. If m fails the test (and is therefore composite), an error results; if it passes the test the Jacobi symbol is computed.

The Jacobi symbol ((n/m)). For odd m > 1 this is defined (but not calculated!) as the product of the Legendre symbols ((n/p_i)), where the product is over all primes p_i dividing m (including multiplicities). Quadratic reciprocity is used to calculate this symbol, which returns -1, 0 or 1.

The Kronecker symbol ((n/m)). This is the extension of the Jacobi symbol to all integers m, by multiplicativity, and by defining ((n/2))=( - 1)^((n^2 - 1)/8) for odd n (and 0 for even n) and ((n/- 1)) equals plus or minus 1 according to the sign of n for n != 0 (and 1 for n = 0).

The Möbius function mu(n). This is a multiplicative function characterized by mu(1)=1, mu(p)= - 1, and mu(p^k)=0 for k >= 2, where p is a prime number.

> d := func< m | DivisorSigma(1, m)-m >;
> z := func< m | d(d(m)) eq m >;
> for m := 2 to 10000 do
>      if z(m) then
>       print m, d(m);
>    end if;
> end for;
6 6
28 28
220 284
284 220
496 496
1184 1210
1210 1184
2620 2924
2924 2620
5020 5564
5564 5020
6232 6368
6368 6232
8128 8128