A103222 -id:A103222 - cp's OEIS frontend

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

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

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}]]

A185952 Partial products of A002313, the primes that are 1 or 2 (mod 4).

Original entry on oeis.org

2, 10, 130, 2210, 64090, 2371330, 97224530, 5152900090, 314326905490, 22945864100770, 2042181904968530, 198091644781947410, 20007256122976688410, 2180790917404459036690, 246429373666703871145970

Offset: 1

Jonathan Vos Post, Feb 07 2011

Product of the first n primes which are natural primes which are not Gaussian primes. Product of the first n primes congruent to 1 or 2 modulo 4. Product of the first n primes of form x^2+y^2. Product of the first n primes p such that -1 is a square mod p. Factors of primorials (A002110) not divisible by natural primes which are also Gaussian primes.

Essentially twice A006278.

			a(10) = 2 * 5 * 13 * 17 * 29 * 37 * 41 * 53 * 61 * 73 = 22945864100770.

G. C. Greubel, Table of n, a(n) for n = 1..300

Cf. A000040, A002110, A002313, A042963, A103222.

Rest@ FoldList[#1*#2 &, 1, Select[ Prime@ Range@ 30, Mod[#, 4] != 3 &]] (* Robert G. Wilson v *)

pp(v)=my(t=1); vector(#v,i,t*=v[i])
pp(select(n->n%4<3, primes(20))) \\ Charles R Greathouse IV, Apr 21 2015

a(n) = Product_{i=1..n} A002313(i) = 2 * Product_{i=1..n} {p in A000040 but p not in A002145} = Product_{i=1..n} {A000040 intersection A042963}.

Terms corrected by Robert G. Wilson v, Feb 11 2011

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 *)

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

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

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A185952 Partial products of A002313, the primes that are 1 or 2 (mod 4).

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

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

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