A102506 -id:A102506 - cp's OEIS frontend

A102507 The imaginary part of a complex multiperfect number, see A102506.

3, 5, 2, 10, 18, 88, 12, 24, 20, 84, 120, 440, 950, 32, 209, 60, 418, 3800, 2256, 768, 120, 1280, 310, 200, 1152, 3840, 600, 4680, 404, 1712, 1240, 6240, 5016

Offset: 1

T. D. Noe, Jan 12 2005

nonn

lst={}; Do[z=n+k*I; s=DivisorSigma[1, z]; If[Mod[s, z]==0, AppendTo[lst, z]; Print[{z, s, s/z}]], {n, 1200}, {k, 10000}]; Im[lst]

A103230 Norm of the sum of divisors function sigma(n) generalized for Gaussian integers.

Original entry on oeis.org

1, 13, 16, 41, 80, 208, 64, 113, 169, 1040, 144, 656, 360, 832, 1280, 481, 520, 2197, 400, 3280, 1024, 1872, 576, 1808, 2257, 4680, 1600, 2624, 1360, 16640, 1024, 2113, 2304, 6760, 5120, 6929, 2000, 5200, 5760, 9040, 2600, 13312, 1936, 5904, 13520

Offset: 1

T. D. Noe, Jan 26 2005

See A102506 for a complete description.
See A103228 and A103229 for the real and imaginary parts.
Multiplicative because the sigma function on Gaussian integers as defined in A102506 is multiplicative and the norm is completely multiplicative. - Andrew Howroyd, Aug 03 2018

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)

Cf. A102506, A102574, A103224, A103228, A103229.

Abs[Table[DivisorSigma[1, n, GaussianIntegers -> True], {n, 100}]]^2

\\ See A102506 for formula.
CSigma(z)={my(f=factor(z,I)); prod(i=1, #f~, my([p,e]=f[i,]); if(norm(p)==1, 1, (p^(e+1)-1)/(p-1)))}
a(n)=norm(CSigma(n)); \\ Andrew Howroyd, Aug 03 2018

a(n) = A103228(n)^2 + A103229(n)^2. - Andrew Howroyd, Aug 03 2018

A103228 Real part of the sum of divisors function sigma(n) generalized for Gaussian integers.

Original entry on oeis.org

1, 2, 4, -4, 4, 8, 8, -8, 13, -16, 12, -16, 6, 16, 16, 16, 6, 26, 20, -56, 32, 24, 24, -32, -36, -42, 40, -32, 8, -64, 32, 32, 48, -54, 32, -52, 8, 40, 24, 24, 10, 64, 44, -48, 52, 48, 48, 64, 57, -165, 24, -114, 10, 80, 48, -64, 80, -92, 60, -224, 12, 64, 104, -64, -120, 96, 68, -134, 96, -128, 72, -104, 12, -116, -144, -80, 96

Offset: 1

T. D. Noe, Jan 26 2005

sign

See A103229 for the imaginary part and A103230 for the norm.

See A102506 for a complete description. Note that sigma(n) is real iff n is in A004614.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A004614, A102506, A103229, A103230.

Re[Table[DivisorSigma[1, n, GaussianIntegers -> True], {n, 100}]]

A103229 Imaginary part of the sum of divisors function sigma(n) generalized for Gaussian integers.

Original entry on oeis.org

0, 3, 0, 5, 8, 12, 0, -7, 0, 28, 0, 20, 18, 24, 32, -15, 22, 39, 0, -12, 0, 36, 0, -28, 31, 54, 0, 40, 36, 112, 0, 33, 0, 62, 64, 65, 44, 60, 72, -92, 50, 96, 0, 60, 104, 72, 0, -60, 0, -46, 88, -42, 62, 120, 96, -56, 0, 96, 0, -48, 72, 96, 0, 65, 120, 144, 0, -58, 0, 224, 0, -91, 84, 112, 124, 100, 0, 216, 0, 68, 0, 130, 0, 160, 136

Offset: 1

T. D. Noe, Jan 26 2005

sign

See A102506 for a complete description.

See A103228 for the real part and A103230 for the norm.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A102506, A103228, A103230.

Im[Table[DivisorSigma[1, n, GaussianIntegers -> True], {n, 100}]]

A102531 Real part of absolute Gaussian perfect numbers, in order of increasing magnitude.

Original entry on oeis.org

3, 15, 6, 19, 111, 91, 159, 72, 472, 904, 2584, 1616, 999, 4328, 702, 4424, 7048, 7328, 2474, 9352, 7144, 7240, 5117, 739, 6327, 15128, 13168, 1263, 14280, 3224, 21704, 15160, 21992, 14044, 23132, 9135, 23656, 24614, 7272, 15464, 9040, 28424, 30956, 14728, 32399

Offset: 1

T. D. Noe, Jan 13 2005

nonn

An absolute Gaussian perfect number z satisfies abs(sigma(z)-z) = abs(z), where sigma(z) is sum of the divisors of z, as defined by Spira for Gaussian integers.

			For z=3+7i, we have sigma(z)-z = 7+3i, which has the same magnitude as z.

Amiram Eldar, Table of n, a(n) for n = 1..76
R. Spira, The Complex Sum Of Divisors, American Mathematical Monthly, 1961 Vol. 68, pp. 120-124.

See A102532 for the imaginary part.

Cf. A102506 and A102507 (Gaussian multiperfect numbers). See also 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}]; Re[Transpose[Sort[lst]][[2]]]

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

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

Original entry on oeis.org

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

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

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A102507 The imaginary part of a complex multiperfect number, see A102506.

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A103230 Norm of the sum of divisors function sigma(n) generalized for Gaussian integers.

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A103228 Real part of the sum of divisors function sigma(n) generalized for Gaussian integers.

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A103229 Imaginary part of the sum of divisors function sigma(n) generalized for Gaussian integers.

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A102531 Real part of absolute Gaussian perfect numbers, in order of increasing magnitude.

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

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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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