A100884 Real parts of multiperfect Gaussian numbers z, sorted with respect to |z| and Re(z).

1, 1, 6, 5, 10, 12, 60, 72, 28, 100, 108, 120, 204, 300, 263, 140, 526, 912, 150, 720, 1470, 1520, 1200, 1704, 672, 600, 4560, 4828, 3600, 5584, 5880, 4680, 6312, 6240, 1800, 2160, 14484, 17640, 8984, 72824, 62400

Offset: 1

Author

Yasutoshi Kohmoto

Keywords

• nonn
• more

Comments

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,...

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.

Examples

			For z = 1, sigma(z) = 1 and m = sigma(z)/z = 1, which adds 1 to the sequence.
For z = 1+3i, sigma(z) = 5+5i and m = sigma(z)/z = 2-i, which adds 1 to the sequence.
For z = 6+2i, sigma(z) = -10+10i and m = sigma(z)/z = -1+2i, which adds 6 to the sequence.
For z = 5+5i, sigma(z) = 20i and m = sigma(z)/ z= 2+2i, which adds 5 to the sequence.
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.
		

Crossrefs

Cf. A100885, A100889, A100890.

Extensions

Entirely rewritten, including the a(n), by R. J. Mathar, Mar 12 2010

a(10)-a(41) from Amiram Eldar, Feb 10 2020