A100890 -id:A100890 - cp's OEIS frontend

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

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

A100889 Real part of the multiplier m of the multiperfect Gaussian integer A100884(n)+i*A100885(n).

Original entry on oeis.org

1, 2, -1, 2, -1, 2, -2, 1, -1, 0, -3, 1, -3, 0, -2, -4, -3, 1, -4, 2, 0, 3, 0, 1, 3, 3, 4, 3, 0, -1, 0, 5, 3, -5, 4, 0, 4, 0, 0, 0, -4

Offset: 1

Yasutoshi Kohmoto

Let the Gaussian integer z = A100884(n)+i*A100885(n). Then a(n) = Re(m) = Re( sigma(z)/z) .

Cf. A100884, A100885, A100890.

Edited, all values replaced by R. J. Mathar, Mar 12 2010

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

A100886 Expansion of x(1+3x+2x^2)/((1+x+x^2)(1-x-x^2)).

Original entry on oeis.org

0, 1, 3, 3, 5, 10, 14, 23, 39, 61, 99, 162, 260, 421, 683, 1103, 1785, 2890, 4674, 7563, 12239, 19801, 32039, 51842, 83880, 135721, 219603, 355323, 574925, 930250, 1505174, 2435423, 3940599, 6376021, 10316619, 16692642, 27009260, 43701901

Offset: 0

Creighton Dement, Nov 21 2004

This sequence was investigated in cooperation with Paul Barry.
Generating floretion: - 0.5'i - 0.5'k - 0.5j' - 0.5'ii' + 0.5'jj' - 0.5'kk' + 0.5'ik' - 0.5'ki' ("tes").
From Joshua P. Bowman, Sep 28 2023: (Start)
a(n) is equal to the number of circular binary sequences of length n+1 with an even number of 0's and no consecutive 1's. A circular binary sequence is a finite sequence of 0's and 1's for which the first and last digits are considered to be adjacent. Rotations are distinguished from each other.
a(n) is also equal to the number of matchings in the cycle graph C_{n+1} for which the number of edges plus the number of unmatched vertices is even.
a(n) is also equal to the number of circular compositions of n+1 into an even number of 1's and 2's. (End)

			When counting circular binary sequences with an even number of 0's and no consecutive 1's, the sequence "1" is not allowed because the 1 is considered to be adjacent to itself. Thus a(0)=0. For n=2, the a(2)=3 allowed sequences of length 3 are 001, 010, and 100. For n=3, the a(3)=3 allowed sequences of length 4 are 0000, 0101, and 1010. - _Joshua P. Bowman_, Sep 28 2023

Wojciech Florek, Table of n, a(n) for n = 0..1001
Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 19.
Wojciech Florek, A class of generalized Tribonacci sequences applied to counting problems, Appl. Math. Comput., 338 (2018), 809-821.
W. O. J. Moser, Cyclic binary strings without long runs of like (alternating) bits, Fibonacci Quart. 31 (1993), no. 1, 2-6.
Index entries for linear recurrences with constant coefficients, signature (0,1,2,1).

Cf. A000032, A007040, A087204, A100887, A100888, A100889, A100890, A366043.

I:=[0,1,3,3]; [n le 4 select I[n] else Self(n-2)+2*Self(n-3)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 30 2015

a[0] = 0; a[1] = 1; a[2] = 3; a[3] = 3; a[n_] := a[n] = a[n - 2] + 2a[n - 3] + a[n - 4]; Table[ a[n], {n, 0, 36}]
(* Or *) CoefficientList[ Series[x(1 + 3x + 2x^2)/((1 + x + x^2)(1 - x - x^2)), {x, 0, 36}], x] (* Robert G. Wilson v, Nov 26 2004 *)
LinearRecurrence[{0,1,2,1},{0,1,3,3},40] (* Harvey P. Dale, Apr 04 2016 *)

a(n):=n*sum(binomial(k,n-k)*(if oddp(k) then 0 else 1/k),k,1,n); /* Vladimir Kruchinin, Apr 09 2011 */

a(n)=n*sum(j=1,n\2,k=2*j;binomial(k,n-k)/k);
vector(66,n,a(n)) /* Joerg Arndt, Apr 09 2011 */

concat([0],Vec(x*(1+3*x+2*x^2)/((1+x+x^2)*(1-x-x^2))+O(x^66))) /* Joerg Arndt, Apr 09 2011 */

(1/2)*(a(n) + A100887(n) - A100888(n)) = A061347(n+3).
a(n) = (L(n+1)-A061347(n+1))/2, L=A000032; [corrected by Wojciech Florek, Feb 26 2018]
a(n) = a(n-2) + 2*a(n-3) + a(n-4), a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 3.
a(n) = n*Sum_{j=1..floor(n/2)} binomial(2*j,n-2*j)/(2*j). - Vladimir Kruchinin, Apr 09 2011 (with offset 1, cf. PARI code)
a(n) = floor(phi^(n+1)/2), n mod 3 = 0,1; a(n) = floor((phi^(n+1)+3)/2), n mod 3 = 2, phi = (1 + sqrt(5))/2; from Binet's formula or the relation to the Lucas numbers A000032. - Wojciech Florek, Mar 03 2018
a(n) = A000032(n+1) - A366043(n+1). - Joshua P. Bowman, Sep 28 2023

More terms from Robert G. Wilson v, Nov 26 2004

A100885 Imaginary part of the Gaussian multiperfect number associated with the real part A100884.

Original entry on oeis.org

0, 3, 2, 5, 10, 18, 12, 24, 88, 20, 84, 120, 32, 60, 209, 440, 418, 120, 950, 768, 310, 200, 1280, 1152, 2256, 3800, 600, 404, 3840, 1712, 1240, 4680, 5016, 6240, 11400, 12880, 1212, 3720, 20012, 1232, 61440

Offset: 1

Yasutoshi Kohmoto

			a(1) = Im(z= 1) with sigma(z) = 1 and m = 1.
a(2) = Im(z= 1+3i) with sigma(z) = 5+5i and m = 2-i.
a(3) = Im(z= 6+2i) with sigma(z) = -10-10i and m = -1+2i.
a(4) = Im(z= 5+5i) with sigma(z) = 20i and m = 2+2i.
a(5) = Im(z= 10+10i) with sigma(z) = -40+20i and m = -1+3i.
a(6) = Im(z= 12+18i) with sigma(z) = -12+60i and m = 2+2i.
a(9) = Im(z= 28+88i) with sigma(z) = -204-32i and m = -1+2i.

Cf. A100884, A100889, A100890.

Edited, all values replaced by R. J. Mathar, Mar 12 2010

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

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 *)

A100887 Expansion of (-1+2x+2x^2)/((1+x+x^2)(1-x-x^2)).

Original entry on oeis.org

-1, 2, 1, 0, 4, 4, 5, 12, 17, 26, 46, 72, 115, 190, 305, 492, 800, 1292, 2089, 3384, 5473, 8854, 14330, 23184, 37511, 60698, 98209, 158904, 257116, 416020, 673133, 1089156, 1762289, 2851442, 4613734, 7465176, 12078907, 19544086, 31622993, 51167076

Offset: 0

Creighton Dement, Nov 21 2004

This sequence was investigated in cooperation with Paul Barry. Generating floretion: - 0.5'i - 0.5'k - 0.5j' - 0.5'ii' + 0.5'jj' - 0.5'kk' + 0.5'ik' - 0.5'ki' ("les").

Index entries for linear recurrences with constant coefficients, signature (0, 1, 2, 1).

Cf. A100886, A100888, A100889, A100890.

a[n_] := Fibonacci[n + 1]/2 - Sqrt[3]Cos[2Pi*n/3 + Pi/6]; Table[ a[n], {n, 0, 39}]
a[0] = -1; a[1] = 2; a[2] = 1; a[3] = 0; a[n_] := a[n] = a[n - 2] + 2a[n - 3] + a[n - 4]; Table[ a[n], {n, 0, 39}]
CoefficientList[ Series[(-1 + 2x + 2x^2)/((1 - x - x^2)(1 + x + x^2)), {x, 0, 39}], x] (* Robert G. Wilson v, Dec 02 2004 *)

Vec((-1+2*x+2*x^2)/((1+x+x^2)*(1-x-x^2))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = Fib(n+1)/2 - sqrt(3)cos(2Pi*n/3 + Pi/6); a(n) = a(n-2) + 2a(n-3) + a(n-4), a(0) = -1, a(1) = 2, a(2) = 1, a(3) = 0

Edited and extended by Robert G. Wilson v, Dec 02 2004

A100888 Expansion of (3+x-x^2)/((1+x+x^2)(1-x-x^2)).

Original entry on oeis.org

3, 1, 2, 7, 7, 12, 23, 33, 54, 91, 143, 232, 379, 609, 986, 1599, 2583, 4180, 6767, 10945, 17710, 28659, 46367, 75024, 121395, 196417, 317810, 514231, 832039, 1346268, 2178311, 3524577, 5702886, 9227467, 14930351, 24157816, 39088171, 63245985

Offset: 0

Creighton Dement, Nov 21 2004

This sequence was investigated in cooperation with Paul Barry. Generating floretion: - 0.5'i - 0.5'k - 0.5j' - 0.5'ii' + 0.5'jj' - 0.5'kk' + 0.5'ik' - 0.5'ki' ("jes"). A100885(n) = (1/2)(A100886(n) + A100887(n) - a(n)).

Cf. A100885, A100886, A100887, A100889, A100890.

a[0] = 3; a[1] = 1; a[2] = 2; a[3] = 7; a[n_] := a[n] = a[n - 2] + 2a[n - 3] + a[n - 4]; Table[ a[n], {n, 0, 37}] (* Robert G. Wilson v, Nov 26 2004 *)
CoefficientList[ Series[(3 + x - x^2)/((1 + x + x^2)(1 - x - x^2)), {x, 0, 37}], x] (* Robert G. Wilson v, Nov 26 2004 *)

Vec((3+x-x^2)/((1+x+x^2)*(1-x-x^2))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = Fib(n+2) + sqrt(3)cos(2Pi*n/3 + Pi/6) + sin(2Pi*n/3 + Pi/6); a(n) = a(n-2) + 2a(n-3) + a(n-4), a(0) = 3, a(1) = 1, a(2) = 2, a(3) = 7.

More terms from Robert G. Wilson v, Nov 26 2004

cp's OEIS Frontend

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

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A100889 Real part of the multiplier m of the multiperfect Gaussian integer A100884(n)+i*A100885(n).

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A100886 Expansion of x*(1+3*x+2*x^2)/((1+x+x^2)*(1-x-x^2)).

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A100885 Imaginary part of the Gaussian multiperfect number associated with the real part A100884.

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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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A100887 Expansion of (-1+2x+2x^2)/((1+x+x^2)(1-x-x^2)).

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A100888 Expansion of (3+x-x^2)/((1+x+x^2)(1-x-x^2)).

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A100886 Expansion of x(1+3x+2x^2)/((1+x+x^2)(1-x-x^2)).