A102531 -id:A102531 - cp's OEIS frontend

A102532 Imaginary part of absolute Gaussian perfect numbers, in order of increasing magnitude.

7, 42, 57, 82, 78, 189, 341, 549, 664, 1048, 1016, 3776, 4072, 1672, 5049, 3816, 1128, 368, 7097, 2504, 6816, 7912, 10743, 12605, 12606, 248, 10392, 16853, 15208, 20824, 344, 15688, 392, 18076, 1340, 22302, 5912, 1573, 24392, 21328, 24896, 7048, 1220, 29288, 9678

Offset: 1

T. D. Noe, Jan 13 2005

nonn

See A102531 for the real part.

Amiram Eldar, Table of n, a(n) for n = 1..76

Cf. A102531, A102506, A102507, A101366, A101367.

lst={}; nn=1000; Do[z=a+b*I; If[Abs[z]<=nn && Abs[(DivisorSigma[1, z]-z)] == Abs[z], AppendTo[lst, {Abs[z]^2, z}]], {a, nn}, {b, nn}]; Im[Transpose[Sort[lst]][[2]]]

a(22)-a(45) from Amiram Eldar, Feb 10 2020

A101366 Perfect Abs: Imaginary part of complex z such that Abs[(Total[Divisors[z]]-z)]=Abs[z].

Original entry on oeis.org

3, 7, 8, 42, 48, 57, 33, 82, 78, 77, 83, 189, 154, 92, 321, 341, 549, 664, 106, 1034, 2929, 4072, 5049, 3037, 6957, 7097, 5051

Offset: 0

Ed Pegg Jr, Jan 13 2005

Having Perfect Abs is not as good as being Perfect. A complex number can also have Abundant Abs or Deficient Abs.

			The divisors for 269+92i are: 1, 2+I, 3+4i, 6+5i, 7+2i, 7+16i, 12+11i, 13+34i, 17+126i, 32+47i, 39+2i, 269+92i. The (sum - k) is 139+248i. Abs[139+248i] == Abs[269+92i]
The sequence of complex z's is: 5+3i, 3+7i, 19+8i, 15+42i, 6+57i, 29+48i, 19+82i, 74+33i, 111+78i, 147+77i, 185+83i, 91+189i, 197+154i, 269+92i, 122+321i, 159+341i, 72+549i, ...

Cf. A101367, A102526, A102531, A102532.

Im[Sort[Select[Flatten[Table[a + b I, {a, 1, 500}, {b, 1, 500}]], Abs[Total[Divisors[ # ]] - # ] == Abs[ # ] &], Abs[ #1] < Abs[ #2] &]]

Ten more terms from Hans Havermann, Jan 15 2005

A101367 Perfect Abs. Real part of complex z such that Abs[(Total[Divisors[z]]-z)]=Abs[z].

Original entry on oeis.org

5, 3, 19, 15, 29, 6, 74, 19, 111, 147, 185, 91, 197, 269, 122, 159, 72, 827, 1487, 2903, 968, 999, 702, 5803, 326, 2474, 7871

Offset: 0

Ed Pegg Jr, Jan 13 2005

nonn

Having Perfect Abs is not as good as being Perfect. A complex number can also have Abundant Abs or Deficient Abs.

			The divisors for 269+92i are: 1, 2+I, 3+4i, 6+5i, 7+2i, 7+16i, 12+11i, 13+34i, 17+126i, 32+47i, 39+2i, 269+92i. The (sum - k) is 139+248i. Abs[139+248i] == Abs[269+92i]

Cf. A101366, A102527, A102531, A102532, A102506, A102507.

Re[Sort[Select[Flatten[Table[a + b I, {a, 1, 500}, {b, 1, 500}]], Abs[Total[Divisors[ # ]] - # ] == Abs[ # ] &], Abs[ #1] < Abs[ #2] &]]

Ten more terms from Hans Havermann, Jan 15 2005

A332318 Real part of Gaussian norm-multiply-perfect numbers, in order of increasing norm.

Original entry on oeis.org

1, 2, 1, 3, 6, 5, 6, 10, 2, 8, 16, 12, 21, 18, 26, 30, 13, 43, 28, 6, 52, 60, 20, 10, 72, 56, 26, 28, 85, 95, 20, 100, 58, 80, 40, 36, 108, 120, 48, 190, 144, 176, 204, 38, 106, 214, 4, 84, 232, 276, 136, 216, 300, 174, 181, 263, 216, 312, 340, 140, 464, 20, 380

Offset: 1

Amiram Eldar, Feb 09 2020

nonn

Norm-multiply-perfect numbers where defined by Spira as Gaussian integers z such that norm(sigma(z)) is divisible by norm(z), where sigma(z) is the sum of the divisors of z = a + b*i and norm(z) = z * conj(z) = a^2 + b^2.
If the ratio norm(sigma(z))/norm(z) is 2 then the number is called a norm-perfect. The norm-perfect numbers correspond to n = 1, 13, 26, ... They are 2 + i, 21 + 22*i, 56 + 64*i, ...
Only nonnegative terms are included, since if z = a + b*i is a norm-multiply-perfect number then z*i, -z and -z*i are also norm-multiply-perfect numbers and one of the four has nonnegative a and b. Terms with the same norm are ordered by their real parts.
The corresponding imaginary parts are in A332319.
The corresponding norms are 1, 5, 10, 10, 40, 50, 52, 200, 260, 260, 260, 468, ...
The corresponding ratios norm(sigma(k))/k are 1, 2, 5, 4, 5, 8, 5, 10, 9, 10, ...

			2 + i is a norm-perfect number since sigma(2 + i) = 3 + i, and (3^2 + 1^2) = 10 = 2 * 5 = 2 * (2^1 + 1^2).
1 + 3*i is a norm-multiply-perfect number since sigma(1 + 3*i) = 5 + 5*i, and (5^2 + 5^2) = 50 = 5 * 10 = 5 * (1^2 + 3^2).

Amiram Eldar, Table of n, a(n) for n = 1..182
Wayne L. McDaniel, Perfect Gaussian integers, Acta Arithmetica, Vol. 2, No. 25 (1974), pp. 137-144.
Robert Spira, The Complex Sum of Divisors, The American Mathematical Monthly, Vol. 68, No. 2 (1961), pp. 120-124.

Cf. A000396, A007691, A100884, A100885, A100889, A100890, A102506, A102507, A102531, A102532, A103228, A103229, A103230, A332319.

csigma[z_] := DivisorSigma[1, z, GaussianIntegers -> True]; normultPerf[z_] := Divisible[Abs[csigma[z]]^2, Abs[z]^2]; seq = {}; max = 10^2; Do[z = a + b*I; If[Abs[z] <= max && normultPerf[z], AppendTo[seq, {Abs[z]^2, z}]], {a, 1, max}, {b, 0, max}]; Re[Transpose[Sort[seq]][[2]]] (* after T. D. Noe at A102531 *)

A332319 Imaginary part of Gaussian norm-multiply-perfect numbers, in order of increasing norm.

Original entry on oeis.org

0, 1, 3, 1, 2, 5, 4, 10, 16, 14, 2, 18, 22, 26, 18, 20, 41, 1, 36, 48, 24, 12, 65, 70, 24, 64, 82, 88, 45, 15, 100, 20, 94, 100, 130, 132, 84, 120, 184, 30, 148, 108, 32, 216, 192, 48, 228, 212, 51, 24, 252, 188, 60, 282, 283, 209, 312, 216, 198, 440, 102, 490

Offset: 1

Amiram Eldar, Feb 09 2020

nonn

See A332318 (the corresponding real parts) for the definition of Gaussian norm-multiply-perfect numbers.

Amiram Eldar, Table of n, a(n) for n = 1..182

Cf. A000396, A007691, A100884, A100885, A100889, A100890, A102506, A102507, A102531, A102532, A103228, A103229, A103230, A332318.

csigma[z_] := DivisorSigma[1, z, GaussianIntegers -> True]; normultPerf[z_] := Divisible[Abs[csigma[z]]^2, Abs[z]^2]; seq = {}; max = 10^2; Do[z = a + b*I; If[Abs[z] <= max && normultPerf[z], AppendTo[seq, {Abs[z]^2, z}]], {a, 1, max}, {b, 0, max}]; Im[Transpose[Sort[seq]][[2]]] (* after T. D. Noe at A102532 *)

A102924 Real part of Gaussian amicable numbers in order of increasing magnitude. See A102925 for the imaginary part.

Original entry on oeis.org

-1105, -1895, -2639, -3235, -3433, -3970, -4694, -3549, -766, -4478, -6880, 5356, -6468, 8008, 4232, -8547

Offset: 1

T. D. Noe, Jan 19 2005

For a Gaussian integer z, let the sum of the proper divisors be denoted by s(z) = sigma(z)-z, where sigma(z) is sum of the divisors of z, as defined by Spira for Gaussian integers. Then z is an amicable Gaussian number if z and s(z) are different and z = s(s(z)). The smallest Gaussian amicable number in the first quadrant is 8008+3960i.

			For z=-1105+1020i, we have s(z)=-2639-1228i and s(s(z))=z.

R. Spira, The Complex Sum Of Divisors, American Mathematical Monthly, 1961 Vol. 68, pp. 120-124.
Eric Weisstein's World of Mathematics, Amicable Pair

Cf. A102506 (Gaussian multiperfect numbers), A102531 (absolute Gaussian perfect numbers).

s[z_Complex] := DivisorSigma[1, z]-z; nn=10000; lst={}; Do[d=a^2+b^2; If[d

cp's OEIS Frontend

A102532 Imaginary part of absolute Gaussian perfect numbers, in order of increasing magnitude.

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A101366 Perfect Abs: Imaginary part of complex z such that Abs[(Total[Divisors[z]]-z)]=Abs[z].

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A101367 Perfect Abs. Real part of complex z such that Abs[(Total[Divisors[z]]-z)]=Abs[z].

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A332318 Real part of Gaussian norm-multiply-perfect numbers, in order of increasing norm.

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A332319 Imaginary part of Gaussian norm-multiply-perfect numbers, in order of increasing norm.

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Mathematica

A102924 Real part of Gaussian amicable numbers in order of increasing magnitude. See A102925 for the imaginary part.

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