A100884 - OEIS

%I #19 Aug 27 2022 04:05:16

%S 1,1,6,5,10,12,60,72,28,100,108,120,204,300,263,140,526,912,150,720,

%T 1470,1520,1200,1704,672,600,4560,4828,3600,5584,5880,4680,6312,6240,

%U 1800,2160,14484,17640,8984,72824,62400

%N Real parts of multiperfect Gaussian numbers z, sorted with respect to |z| and Re(z).

%C Sort the Gaussian integers z in the first quadrant according to increasing modulus |z|, and within the same modulus according to increasing Re(z): 1, 1+i, 2, 1+2i, 2+i, 2+2i, 3, 1+3i, 3+i, 2+3i, 3+2i,...

%C If z divides the value of sigma(z), defined in A103228, i.e., if sigma(z)=z*m with m some Gaussian integer (m not necessarily in the first quadrant), add Re(z) to the sequence.

%e For z = 1, sigma(z) = 1 and m = sigma(z)/z = 1, which adds 1 to the sequence.

%e For z = 1+3i, sigma(z) = 5+5i and m = sigma(z)/z = 2-i, which adds 1 to the sequence.

%e For z = 6+2i, sigma(z) = -10+10i and m = sigma(z)/z = -1+2i, which adds 6 to the sequence.

%e For z = 5+5i, sigma(z) = 20i and m = sigma(z)/ z= 2+2i, which adds 5 to the sequence.

%e For z = (1+i)^7 = 8-8i, the divisors are 1, 1+i, (1+i)^2 = 2i, (1+i)^3 = -2+2i, (1+i)^4 = -4, (1+i)^5= -4-4i, (1+i)^6 = -8i, (1+i)^7 = 8-8i. So sigma(z) is 1 +1+i +2i -2+2i -4 -4-4i -8i +8-8i = -15i and sigma(z)/z is m = -15i/(8-8i) which is not a Gaussian integer, so Re(z)=8 is NOT added to the sequence.

%Y Cf. A100885, A100889, A100890.

%K nonn,more

%O 1,3

%A _Yasutoshi Kohmoto_

%E Entirely rewritten, including the a(n), by _R. J. Mathar_, Mar 12 2010

%E a(10)-a(41) from _Amiram Eldar_, Feb 10 2020