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.