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%I #9 Feb 03 2017 10:47:27
%S 1,5,6,10,12,28,60,72,100,108,120,140,150,204,263,300,526,600,672,720,
%T 912,1200,1470,1520,1704,3600,4560,4680,4828,5584,5880,6240,6312
%N Numbers n such that for some positive number k, z=n+ik is a complex multiperfect number; that is, z divides sigma(z), where sigma is the sum of divisors function extended to the complex numbers.
%C 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.
%C Then Spira defines the sum of divisors to be prod_{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.)
%C 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?
%H R. Spira, <a href="http://www.jstor.org/stable/2312472">The Complex Sum Of Divisors</a>, American Mathematical Monthly, 1961 Vol. 68, pp. 120-124.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MultiperfectNumber.html">Multiperfect Number</a>
%e For n=1, we have z=1+3i. The divisors of z are 1, 1+i, 1+3i and 2+i. Hence sigma(z)=5+5i and sigma(z)/z = 2-i.
%t lst={}; Do[z=n+k*I; s=DivisorSigma[1, z]; If[Mod[s, z]==0, AppendTo[lst, z]; Print[{z, s, s/z}]], {n, 1200}, {k, 10000}]; Re[lst]
%Y Cf. A102507. Note that A101367 and A101366 use Mathematica's Divisors function, the sum of the first-quadrant divisors, which does not enjoy the nice multiplicative properties of Spira's sigma function.
%K nice,nonn,more
%O 1,2
%A _T. D. Noe_, Jan 12 2005
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