A103224 -id:A103224 - cp's OEIS frontend

A103222 Real part of the totient function phi(n) for Gaussian integers. See A103223 for the imaginary part and A103224 for the norm.

1, 1, 2, 2, 2, 2, 6, 4, 6, 0, 10, 4, 8, 6, 4, 8, 12, 6, 18, 0, 12, 10, 22, 8, 10, 4, 18, 12, 22, 0, 30, 16, 20, 8, 12, 12, 30, 18, 16, 0, 32, 12, 42, 20, 12, 22, 46, 16, 42, 0, 24, 8, 44, 18, 20, 24, 36, 16, 58, 0, 50, 30, 36, 32, 8, 20, 66, 16, 44, 0, 70, 24, 62, 24, 20, 36, 60, 8, 78, 0

Offset: 1

T. D. Noe, Jan 26 2005

This definition of the totient function for Gaussian integers preserves many of the properties of the usual totient function: (1) it is multiplicative: if gcd(z1,z2)=1, then phi(z1*z2)=phi(z1)*phi(z2), (2) phi(z^2)=z*phi(z), (3) z=Sum_{d|z} phi(d) for properly selected divisors d and (4) the congruence z=1 (mod phi(z)) appears to be true only for Gaussian primes. The first negative term occurs for n=130=2*5*13, the product of the first three primes which are not Gaussian primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Eric Weisstein's World of Mathematics, Totient Function

Cf. A103223, A103224.

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]]; Re[Table[phi[n], {n, 100}]]

Let a nonzero Gaussian integer z have the factorization u p1^e1...pn^en, where u is a unit (1, i, -1, -i), the pk are Gaussian primes in the first quadrant and the ek positive integers. Then we define phi(z) = u*product_{k=1..n} (pk-1) pk^(ek-1).

A103223 Imaginary part of the totient function phi(n) for Gaussian integers. See A103222 for the real part and A103224 for the norm.

Original entry on oeis.org

0, 1, 0, 2, 2, 2, 0, 4, 0, 4, 0, 4, 4, 6, 4, 8, 4, 6, 0, 8, 0, 10, 0, 8, 10, 12, 0, 12, 6, 8, 0, 16, 0, 16, 12, 12, 6, 18, 8, 16, 8, 12, 0, 20, 12, 22, 0, 16, 0, 20, 8, 24, 8, 18, 20, 24, 0, 28, 0, 16, 10, 30, 0, 32, 24, 20, 0, 32, 0, 24, 0, 24, 10, 36, 20, 36, 0, 24, 0, 32, 0, 40, 0, 24, 32, 42

Offset: 1

T. D. Noe, Jan 26 2005

nonn

Note that a(n)=0 when n is in A004614, the product of real Gaussian primes. It appears that all terms are nonnegative.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A103222, A103224.

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]]; Im[Table[phi[n], {n, 100}]]

A332316 Numbers k such that k and k + 1 have the same value of the norm of the totient function for Gaussian integers (A103224).

Original entry on oeis.org

4, 5, 35, 51, 154, 804, 3596, 6200, 7595, 916538, 1638039, 2794805, 6804035, 24724472, 40128444, 52424787, 69918849, 82954611, 124077316, 160245605, 204215514, 361275551, 371254235, 661831521, 1314759754, 1695554762, 2246110022, 2378746131, 2889320799, 4181707719

Offset: 1

Amiram Eldar, Feb 09 2020

nonn

The corresponding common values A103224(k) = A103224(k+1) are 8, 8, 288, 640, 7200, 139392, 3744000, 5760000, ...

			4 is a term since norm(phi(4)) = norm(phi(5)) = 8.

Amiram Eldar, Table of n, a(n) for n = 1..37

Cf. A001274, A103222, A103223, A103224, A287055, A293184, A326403.

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]]; Select[Range[10^3], Abs[phi[#]]^2 == Abs[phi[# + 1]]^2 &] (* after T. D. Noe at A103222 *)

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

A103225 Number of Gaussian integers z with abs(z) < n and gcd(n,z)=1.

Original entry on oeis.org

1, 4, 24, 24, 44, 48, 144, 96, 224, 96, 372, 192, 444, 304, 404, 392, 792, 448, 1124, 408, 1200, 752, 1648, 808, 1240, 896, 2036, 1200, 2440, 800, 2996, 1600, 3008, 1592, 2404, 1808, 4056, 2256, 3616, 1600, 4992, 2400, 5784, 3008, 3604, 3304, 6916, 3224, 7376

Offset: 1

T. D. Noe, Jan 26 2005

This sequence is much like the usual totient function. That is, it gives the number of Gaussian integers that are relatively prime to n and whose modulus is less than n. When n is a Gaussian prime, A002145, then a(n) = A051132(n)-1.

Four of the dominant lines of the plot appear to align to k(i)*Pi*n^2, with k(i) = 1, 8/9, 1/2, and 4/9. Conjecture: a(n) < Pi*n^2. - Bill McEachen, Aug 14 2025

			a(2)=4 because 1, -1, i and -i are relatively prime to 2 and have modulus less than 2.

Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..200 from T. D. Noe)

Cf. A002145, A051132, A103222, A103223, A103224.

Table[cnt=0; Do[z=a+ b*I; If[Abs[z]

cp's OEIS Frontend

A103222 Real part of the totient function phi(n) for Gaussian integers. See A103223 for the imaginary part and A103224 for the norm.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A103223 Imaginary part of the totient function phi(n) for Gaussian integers. See A103222 for the real part and A103224 for the norm.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A332316 Numbers k such that k and k + 1 have the same value of the norm of the totient function for Gaussian integers (A103224).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A103225 Number of Gaussian integers z with abs(z) < n and gcd(n,z)=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica