A103230 -id:A103230 - cp's OEIS frontend

A332320 Numbers k that are highly norm-abundant in Gaussian integers, i.e., A103230(m) < A103230(k) for all m < k.

1, 2, 3, 4, 5, 6, 10, 15, 18, 20, 26, 30, 50, 60, 70, 78, 90, 130, 150, 170, 180, 210, 260, 270, 330, 390, 510, 630, 780, 870, 910, 990, 1020, 1050, 1110, 1170, 1530, 1890, 1950, 2210, 2340, 2550, 2730, 3510, 4290, 4590, 5070, 5460, 5610, 5850, 6630, 8190, 10530

Offset: 1

Amiram Eldar, Feb 09 2020

nonn

Analogous to highly abundant numbers (A002093), with the norm of the sum of divisors function generalized for Gaussian integers (A103230) instead of the sum of divisors function (A000203).

			The first 6 terms of A103230 are 1, 13, 16, 41, 80, 208, 64, 113, 169, 1040. The record values occur at n = 1, 2, 3, 4, 5, 6, 10, the first 7 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..375
Robert Spira, The Complex Sum of Divisors, The American Mathematical Monthly, Vol. 68, No. 2 (1961), pp. 120-124.

Cf. A000203, A002093, A103228, A103229, A103230, A279254, A332321.

s[n_] := Abs[DivisorSigma[1, n, GaussianIntegers -> True]]^2; sm = 0; seq = {}; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1,10^4}]; seq

A332321 Numbers k that are norm-superabundant in Gaussian integers, i.e., A103230(m)/m^2 < A103230(k)/k^2 for all m < k.

Original entry on oeis.org

1, 2, 6, 10, 30, 90, 130, 210, 390, 1170, 2730, 5850, 6630, 19890, 46410, 99450, 139230, 192270, 576810, 1345890, 2884050, 4037670, 7883070, 12113010, 20188350, 23649210, 44414370, 49797930, 55181490, 118246050, 149393790, 165544470, 496633410, 746968950, 827722350

Offset: 1

Amiram Eldar, Feb 09 2020

nonn

Analogous to superabundant numbers (A004394), with the magnitude of the sum of divisors function generalized for Gaussian integers (sqrt(A103230)) instead of the sum of divisors function (A000203).

			The first 6 terms of A103230 are 1, 13, 16, 41, 80, 208. The corresponding values of A103230(n)/n^2 are 1, 3.25, 1.777..., 2.5625, 3.2, 5.777... and the record values occur at n = 1, 2, 6, the first 3 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..40
Robert Spira, The Complex Sum of Divisors, The American Mathematical Monthly, Vol. 68, No. 2 (1961), pp. 120-124.

Cf. A000203, A004394, A103228, A103229, A103230, A279254, A332320.

r[n_] := Abs[DivisorSigma[1, n, GaussianIntegers -> True]]^2/n^2; rm = 0; seq = {}; Do[r1 = r[n]; If[r1 > rm, rm = r1; AppendTo[seq, n]], {n, 1, 6*10^5}]; seq

A332531 Even numbers k such that A103230(k) is a perfect square.

Original entry on oeis.org

442, 818, 1130, 1226, 1326, 1576, 2454, 3094, 3390, 3678, 3978, 4728, 4862, 5330, 5726, 5986, 6452, 7362, 7786, 7910, 8362, 8398, 8582, 8998, 9282, 10166, 10170, 10250, 11032, 11034, 11934, 12410, 12430, 13486, 13702, 14184, 14586, 15542, 15990, 17178, 17336

Offset: 1

Amiram Eldar, Feb 15 2020

nonn

The odd numbers k such that A103230(k) is a perfect square are the numbers that are divisible only by primes congruent to 3 mod 4 (A004614).

			442 is a term since A103230(442) = 2433600 = 1560^2.

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

Cf. A004614, A006532, A103230.

Select[2 * Range[9000], IntegerQ @ Sqrt[Abs[DivisorSigma[1, #, GaussianIntegers -> True]]^2] &]

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}]]

A103224 Norm of the totient function phi(n) for Gaussian integers. See A103222 and A103223 for the real and imaginary parts.

Original entry on oeis.org

1, 2, 4, 8, 8, 8, 36, 32, 36, 16, 100, 32, 80, 72, 32, 128, 160, 72, 324, 64, 144, 200, 484, 128, 200, 160, 324, 288, 520, 64, 900, 512, 400, 320, 288, 288, 936, 648, 320, 256, 1088, 288, 1764, 800, 288, 968, 2116, 512, 1764, 400, 640, 640, 2000, 648, 800, 1152

Offset: 1

T. D. Noe, Jan 26 2005

See A103222 for definitions.

Multiplicative because the totient function on Gaussian integers 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 T. D. Noe)

Cf. A103222, A103223, A103230.

phi[z_] := Module[{f, k, prod}, If[Abs[z]==1, z, f=FactorInteger[z, GaussianIntegers->True]; If[Abs[f[[1, 1]]]==1, k=2; prod=f[[1, 1]], k=1; prod=1]; Do[prod=prod*(f[[i, 1]]-1)f[[i, 1]]^(f[[i, 2]]-1), {i, k, Length[f]}]; prod]]; Abs[Table[phi[n], {n, 100}]]^2

\\ See A103222
CEulerPhi(z)={my(f=factor(z,I)); prod(i=1, #f~, my([p,e]=f[i,]); if(norm(p)==1, p^e, (p-1)*p^(e-1)))}
a(n)=norm(CEulerPhi(n)); \\ Andrew Howroyd, Aug 03 2018

a(n) = A103222(n)^2 + A103223(n)^2. - Andrew Howroyd, Aug 03 2018

A332474 The norm of the sum of unitary divisors function (usigma) generalized for Gaussian integers.

Original entry on oeis.org

1, 5, 16, 9, 80, 80, 64, 65, 100, 400, 144, 144, 360, 320, 1280, 289, 520, 500, 400, 720, 1024, 720, 576, 1040, 640, 1800, 784, 576, 1360, 6400, 1024, 1025, 2304, 2600, 5120, 900, 2000, 2000, 5760, 5200, 2600, 5120, 1936, 1296, 8000, 2880, 2304, 4624, 2500, 3200

Offset: 1

Amiram Eldar, Feb 13 2020

nonn

See A332472 for a description.

			a(4) = 9 since 4 = -(1 + i)^4 in Gaussian integers (i is the imaginary unit), so usigma(4) = (1 + i)^4 + 1 = -3, and a(4) = (-3)^2 + 0^2 = 9.

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

Cf. A034448, A103230, A332472 (the real part), A332473 (the imaginary part).

f[p_, e_] := If[Abs[p] == 1, 1, (p^e + 1)]; usigma[n_] := Times @@ f @@@ FactorInteger[n, GaussianIntegers -> True]; a[n_] := Abs[usigma[n]]^2; Array[a, 100]

a(n) = A332472(n)^2 + A332473(n)^2.

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 *)

A332570 Numbers that are norm-abundant in Gaussian integers.

Original entry on oeis.org

2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 58, 60, 62, 63, 64, 65, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 94, 95, 96

Offset: 1

Amiram Eldar, Feb 16 2020

nonn

Numbers k such that N(sigma(k)) > 2*N(k) = 2*k^2, where sigma(k) = A103228(k) + i*A103229(k) is the sum of divisors of k in Gaussian integers (i is the imaginary unit), and N(z) = Re(z)^2 + Im(z)^2 is the norm of the complex number z.

The number of terms not exceeding 10^k for k = 1, 2, ... is 6, 70, 711, 7002, 69925, 701081, 7016287, 70074003, 700557394, 7007078826, ... Apparently this sequence has an asymptotic density of ~0.7.

			2 is norm-abundant since sigma(2) = 2 + 3*i and N(2 + 3*i) = 2^2 + 3^2 = 13 > 2 * 2^2 = 8.

Miriam Hausman, On Norm Abundant Gaussian Integers, The Journal of the Indian Mathematical Society, Vol. 49 (1987), pp. 119-123.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter III, page 120.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Robert Spira, The Complex Sum of Divisors, The American Mathematical Monthly, Vol. 68, No. 2 (1961), pp. 120-124.

Cf. A005101, A103228, A103229, A103230, A332320, A332321, A332571, A332572.

normAbQ[z_] := Abs[DivisorSigma[1, z, GaussianIntegers -> True]]^2 > 2*Abs[z]^2; Select[Range[100], normAbQ]

cp's OEIS Frontend

A332320 Numbers k that are highly norm-abundant in Gaussian integers, i.e., A103230(m) < A103230(k) for all m < k.

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A332321 Numbers k that are norm-superabundant in Gaussian integers, i.e., A103230(m)/m^2 < A103230(k)/k^2 for all m < k.

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A332531 Even numbers k such that A103230(k) is a perfect square.

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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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A103224 Norm of the totient function phi(n) for Gaussian integers. See A103222 and A103223 for the real and imaginary parts.

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A332474 The norm of the sum of unitary divisors function (usigma) generalized for Gaussian integers.

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

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