A353151 A Gaussian integer analog of the sum-of-divisors function (see Comments lines for definition).

1, 5, 4, 13, 10, 20, 8, 25, 13, 50, 12, 52, 20, 40, 40, 41, 26, 65, 20, 130, 32, 60, 24, 100, 61, 100, 40, 104, 40, 200, 32, 65, 48, 130, 80, 169, 50, 100, 80, 250, 52, 160, 44, 156, 130, 120, 48, 164, 57, 305, 104, 260, 68, 200, 120, 200, 80, 200, 60, 520, 74, 160

Offset: 1

Author

Allan C. Wechsler, Apr 26 2022

Keywords

• nonn
• mult

Comments

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).

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

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.

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.

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.

Examples

			2 = (1+i)(1-i), so a(2) = (2+i)(2-i) = 5.
3 is already a Gaussian prime, so a(3) = 1 + 3 = 4.
4 = (1+i)^2 (1-i)^2, so a(4) = (1 + (1+i) + (1+i)^2) (1 + (1-i) + (1-i)^2)
  = (2+3i)(2-3i) = 13.
12 = 2^2 * 3, so by real multiplicativity (see comments), a(12) = 13 * 4 = 52.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A000404, A317797.

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

Formula

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.)

Extensions

More terms from David A. Corneth, Apr 27 2022