A103228 -id:A103228 - cp's OEIS frontend

A332317 Numbers k that are harmonic in Gaussian integers: k * A062327(k) is divisible by A103228(k) + i*A103229(k) (where i is the imaginary unit).

1, 5, 130, 390, 585, 3250, 31980, 133250, 223860, 799500, 7195500, 13591500, 122323500, 258238500, 394153500, 405346500, 910630500, 1345558500, 2025133500, 8195674500

Offset: 1

Amiram Eldar, Feb 09 2020

Analogous to harmonic numbers (A001599), with the number and sum of divisors functions generalized for Gaussian integers (A062327, A103228, A103229) instead of the number and sum of divisors functions (A000005, A000203).

No more terms below 10^10.

			5 is a term since 5 * A062327(5)/(A103228(5) + i*A103229(5)) = 5 * 4 /(4 + 8*i) = 1 - 2*i is a Gaussian integer.

Cf. A000005, A000203, A001599, A062327, A103228, A103229, A103230.

Select[Range[10^4], Divisible[# * DivisorSigma[0, #, GaussianIntegers -> True], DivisorSigma[1, #, GaussianIntegers -> True]] &]

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

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}]]

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

Original entry on oeis.org

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

Yasutoshi Kohmoto

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.

			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.

Cf. A100885, A100889, A100890.

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

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

A332472 The real part of the sum of unitary divisors function (usigma) generalized for Gaussian integers.

Original entry on oeis.org

1, 1, 4, -3, 4, 4, 8, 1, 10, -12, 12, -12, 6, 8, 16, 17, 6, 10, 20, -12, 32, 12, 24, 4, -24, -30, 28, -24, 8, -48, 32, 1, 48, -38, 32, -30, 8, 20, 24, 68, 10, 32, 44, -36, 40, 24, 48, 68, 50, -40, 24, -18, 10, 28, 48, 8, 80, -64, 60, -48, 12, 32, 80, -63, -120

Offset: 1

Amiram Eldar, Feb 13 2020

sign

If n = u * Product_{i} p_i^e_i, where u is a unit (1, i, -1 or -i), and p_i is a Gaussian prime with Re(p_i) > 0, then usigma(n) = Product_{i} (p_i^e_i + 1).

a(n) = A103228(n) for odd squarefree numbers (A056911), i.e., numbers n such that A318608(n) != 0.

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

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

Cf. A034448, A103228, A332473 (the imaginary part), A332474 (the norm).

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

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]

A332572 Numbers that are norm-deficient in Gaussian integers.

Original entry on oeis.org

1, 3, 7, 8, 11, 16, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 57, 59, 61, 67, 69, 71, 73, 77, 79, 83, 89, 93, 97, 101, 103, 107, 109, 113, 121, 127, 128, 129, 131, 133, 137, 139, 141, 149, 151, 152, 157, 161, 163, 167, 173, 177, 179, 181, 184, 191, 193, 197

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 4, 30, 289, 2998, 30075, 298919, 2983713, 29925997, 299442606, 2992921174, ... Apparently this sequence has an asymptotic density of ~0.3.

			3 is norm-deficient since sigma(3) = 4 and N(4) = 4^2 = 16 < 2 * 3^2 = 18.
8 is norm-deficient since sigma(8) = -8 - 7*i and N(-8 - 7*i) = (-8)^2 + (-7)^2 = 113 < 2 * 8^2 = 128.

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

Cf. A005100, A103228, A103229, A103230, A332570.

normDefQ[z_] := Abs[DivisorSigma[1, z, GaussianIntegers -> True]]^2 < 2*Abs[z]^2; Select[Range[200], normDefQ]

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

Original entry on oeis.org

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

cp's OEIS Frontend

A332317 Numbers k that are harmonic in Gaussian integers: k * A062327(k) is divisible by A103228(k) + i*A103229(k) (where i is the imaginary unit).

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A103230 Norm 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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A100884 Real parts of multiperfect Gaussian numbers z, sorted with respect to |z| and Re(z).

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A332472 The real part 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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A332319 Imaginary part of Gaussian norm-multiply-perfect numbers, in order of increasing norm.

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A332570 Numbers that are norm-abundant in Gaussian integers.

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A332572 Numbers that are norm-deficient in Gaussian integers.

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