A353151 - OEIS

%I #23 Apr 28 2022 13:31:47

%S 1,5,4,13,10,20,8,25,13,50,12,52,20,40,40,41,26,65,20,130,32,60,24,

%T 100,61,100,40,104,40,200,32,65,48,130,80,169,50,100,80,250,52,160,44,

%U 156,130,120,48,164,57,305,104,260,68,200,120,200,80,200,60,520,74,160

%N A Gaussian integer analog of the sum-of-divisors function (see Comments lines for definition).

%C 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).

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

%C 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.

%C 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.

%C 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.

%H David A. Corneth, <a href="/A353151/b353151.txt">Table of n, a(n) for n = 1..10000</a>

%F Factorize n over the Gaussian integers into the form Product (p(i)^e(i)), where Re(p(i)) >= Im(p(i)). Then a(n) = Product (p(i)^(e(i)+1) - 1)/(p(i) - 1). (This has no imaginary part since it is a product of conjugate pairs.)

%e 2 = (1+i)(1-i), so a(2) = (2+i)(2-i) = 5.

%e 3 is already a Gaussian prime, so a(3) = 1 + 3 = 4.

%e 4 = (1+i)^2 (1-i)^2, so a(4) = (1 + (1+i) + (1+i)^2) (1 + (1-i) + (1-i)^2)

%e   = (2+3i)(2-3i) = 13.

%e 12 = 2^2 * 3, so by real multiplicativity (see comments), a(12) = 13 * 4 = 52.

%Y Cf. A000203, A000404, A317797.

%Y Analogic multiperfect numbers under a similar interpretation of sum of complex divisors: A102506, A102507.

%K nonn,mult

%O 1,2

%A _Allan C. Wechsler_, Apr 26 2022

%E More terms from _David A. Corneth_, Apr 27 2022