A391110: misleading definition and other problems.
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Miles Englezou
4:33 AM (7 hours ago) 4:33 AM
to SeqFan
Dear all,
I propose making quite a few changes to A391110. Since there are so many, I've posted here to describe them, and I will refer to this post in a pink box comment when editing the sequence.
A391110 has the name "Euler's totient function on Gaussian integers: number of elements in a reduced system of residues modulo a + b*sqrt(-1)"
However, upon examining the sequence, you'll find that it is actually describing this:
Let phi_i : Z[i] -> Z be the analog of Euler's totient function over the Gaussian integers. Then a(n) = phi_i(x + yi) where x^2 + y^2 = A001481(n), the n-th sum of two squares, for x and y such that x <= y and (y - x) > (s - r) for all other r and s such that r^2 + s^2 = A001481(n), r <= s.
So, despite what the name suggests, it does not give the full multiset of Gaussian totient values but a proper subset of this multiset. For example, a(14) = 16 because A001481(14) = 25 = 0^2 + 25^2 = 3^2 + 4^2, and phi_i(0 + 25i) = 16. However phi_i(3 + 4i) = 20. And though 20 happens to be a term because it is produced by another pair x,y, there are numbers which are Gaussian totient values which are not included in the sequence, numbers like 500, which is produced by phi_i(7 + 24i); now, 7^2 + 24^2 = 625 = A001481(216), but here a(216) = 400 = phi_i(0 + 25i).
Apart from the misleading name there are several things not right with it.
- It lists 0 as a term, however that does not make sense in light of the name, for there is no reduced residue system of 0 elements.
- The comment obfuscates the definition of the sequence behind A-numbers hidden within a general and borderline irrelevant discussion of how to compute Gaussian totient values. It also states "This sequence is the canonical generalization of the Fermat-Euler theorem to imaginary quadratic field Q(i)." which is misleading.
- It has as a formula a(n) = phi(A229140(n) + A385236(n)*i). Which does not make sense if we interpret phi as Euler's totient function, since of course that is only defined on rational integers. If the formula means phi to represent the generalisation of Euler's phi to the Gaussian integers, then this is not indicated.
There are several more cosmetic issues, like use of a as a variable, and a Pari program that's hard-coded to simply print the first 173 terms.
I propose to
- Edit the name to reflect its truer, more restricted definition; or at least to point to a comment defining x and y. Remove entirely "Euler's totient function on Gaussian integers"
- Add a comment explicitly defining the sequence as I defined it above.
- Edit the existing comment to remove misleading definitions.
- Remove 0 as a term by changing the offset from 1 to 2.
- Change phi in the formula to phi_i or something similar, and pointing to its definition in the comments.
- Change variables a and b to x and y.
- Lightly edit the Pari so that it gives a list of k terms, instead of a hard-coded 173 terms as it currently does.
Kind regards,
Miles
I propose making quite a few changes to A391110. Since there are so many, I've posted here to describe them, and I will refer to this post in a pink box comment when editing the sequence.
A391110 has the name "Euler's totient function on Gaussian integers: number of elements in a reduced system of residues modulo a + b*sqrt(-1)"
However, upon examining the sequence, you'll find that it is actually describing this:
Let phi_i : Z[i] -> Z be the analog of Euler's totient function over the Gaussian integers. Then a(n) = phi_i(x + yi) where x^2 + y^2 = A001481(n), the n-th sum of two squares, for x and y such that x <= y and (y - x) > (s - r) for all other r and s such that r^2 + s^2 = A001481(n), r <= s.
So, despite what the name suggests, it does not give the full multiset of Gaussian totient values but a proper subset of this multiset. For example, a(14) = 16 because A001481(14) = 25 = 0^2 + 25^2 = 3^2 + 4^2, and phi_i(0 + 25i) = 16. However phi_i(3 + 4i) = 20. And though 20 happens to be a term because it is produced by another pair x,y, there are numbers which are Gaussian totient values which are not included in the sequence, numbers like 500, which is produced by phi_i(7 + 24i); now, 7^2 + 24^2 = 625 = A001481(216), but here a(216) = 400 = phi_i(0 + 25i).
Apart from the misleading name there are several things not right with it.
- It lists 0 as a term, however that does not make sense in light of the name, for there is no reduced residue system of 0 elements.
- The comment obfuscates the definition of the sequence behind A-numbers hidden within a general and borderline irrelevant discussion of how to compute Gaussian totient values. It also states "This sequence is the canonical generalization of the Fermat-Euler theorem to imaginary quadratic field Q(i)." which is misleading.
- It has as a formula a(n) = phi(A229140(n) + A385236(n)*i). Which does not make sense if we interpret phi as Euler's totient function, since of course that is only defined on rational integers. If the formula means phi to represent the generalisation of Euler's phi to the Gaussian integers, then this is not indicated.
There are several more cosmetic issues, like use of a as a variable, and a Pari program that's hard-coded to simply print the first 173 terms.
I propose to
- Edit the name to reflect its truer, more restricted definition; or at least to point to a comment defining x and y. Remove entirely "Euler's totient function on Gaussian integers"
- Add a comment explicitly defining the sequence as I defined it above.
- Edit the existing comment to remove misleading definitions.
- Remove 0 as a term by changing the offset from 1 to 2.
- Change phi in the formula to phi_i or something similar, and pointing to its definition in the comments.
- Change variables a and b to x and y.
- Lightly edit the Pari so that it gives a list of k terms, instead of a hard-coded 173 terms as it currently does.
Kind regards,
Miles
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