cp's OEIS Frontend

This sequence uses a number-theoretic extension of the sigma function that is due to Spira. A nonzero Gaussian integer has a unique factorization as u q1^e1 q2^e2..qn^en, where u is a unit (1,-1,i,-i), the qk are Gaussian primes in the first quadrant and the ek are positive integers.

Then Spira defines the sum of divisors to be Product_{k=1..n} (qk^(ek+1)-1)/(qk-1). This appears to be the natural number-theoretic extension. Spira's definition preserves the multiplicative property: if GCD(x,y)=1, then sigma(x*y)=sigma(x)*sigma(y). (Mathematica's DivisorSigma function uses this formula.)

It appears that the value of k, A102507, is unique for each n. The sum of divisors function, as defined by Spira, is implemented in Mathematica for complex z as the DivisorSigma[1,z]. For the z=n+ik given here, sigma(z)/z is usually a small Gaussian integer. The first instance of a positive integral value of sigma(z)/z is z=600+3800i, in which case the ratio is 3. The complex multiperfect numbers can be arranged into classes according to the value of sigma(z)/z. Does each class have a finite number of members?

An absolute Gaussian perfect number z satisfies abs(sigma(z)-z) = abs(z), where sigma(z) is sum of the divisors of z, as defined by Spira for Gaussian integers.

See A102531 for the real part.

Having Perfect Abs is not as good as being Perfect. A complex number can also have Abundant Abs or Deficient Abs.

Norm-multiply-perfect numbers where defined by Spira as Gaussian integers z such that norm(sigma(z)) is divisible by norm(z), where sigma(z) is the sum of the divisors of z = a + b*i and norm(z) = z * conj(z) = a^2 + b^2.

If the ratio norm(sigma(z))/norm(z) is 2 then the number is called a norm-perfect. The norm-perfect numbers correspond to n = 1, 13, 26, ... They are 2 + i, 21 + 22*i, 56 + 64*i, ...

Only nonnegative terms are included, since if z = a + b*i is a norm-multiply-perfect number then z*i, -z and -z*i are also norm-multiply-perfect numbers and one of the four has nonnegative a and b. Terms with the same norm are ordered by their real parts.

The corresponding imaginary parts are in A332319.

The corresponding norms are 1, 5, 10, 10, 40, 50, 52, 200, 260, 260, 260, 468, ...

The corresponding ratios norm(sigma(k))/k are 1, 2, 5, 4, 5, 8, 5, 10, 9, 10, ...

See A332318 (the corresponding real parts) for the definition of Gaussian norm-multiply-perfect numbers.

Definition: Multiplicative over the Gaussian integers. Factorize n into Gaussian prime factors whose imaginary part does not exceed their real part. Then, for each distinct Gaussian prime power factor p^k, calculate (1 + p + ... + p^k) = (p^(k+1) - 1) / (p - 1) ; multiply all these Gaussian prime power contributions to get a(n).

It is not clear if this is the same as Spira's complex sum-of-divisors function; see A102506.

This is a Gaussian sum of divisors function, in that it is a sum of one associate of each Gaussian divisor of n; it's just not clear that we choose the same associate as Spira does in all cases.

If m and n are relatively prime real integers, then they are relatively prime Gaussian integers, so this function is also multiplicative in the usual sense, over the real integers.

Note that under this sum-of-divisors function, 5 is analogically perfect, and 10 is analogically multiperfect with index 5, because a(5) = 10, and a(10) = 50.