Arithmetic functions should be methods of Sage integers

[kedlaya]

One common use model of Sage is to use tab completion to identify methods of a given object. For Sage integers, this will currently fail to reveal many standard functions such as the following.

number_of_divisors # d(n)
moebius            # mu(n)
euler_phi          # phi(n)
partitions         # tau(n)

I propose to add methods to the class of Sage integers. In the process, perhaps one can add a few standard functions which are missing even from the class of bare functions, like sum_of_divisors ("sigma(n)").

CC: @kcrisman

Component: number theory

Keywords: arithmetic functions

Issue created by migration from https://trac.sagemath.org/ticket/14401

[rwst]

comment:3

I doubt the general usefulness of such an undertaking. The notion that functions that take an object as parameter should be equivalent to member functions of that object is too broad.

[kedlaya]

comment:7

I agree that the notion is too broad, but not in the way you have in mind: in Python, one generally tries to avoid clogging up the global namespace with things that really should be member functions of a particular class. So if I were starting from scratch, I would strongly consider implementing these functions exclusively as member functions of Sage integers; but I don't think it's appropriate to move in that direction now.

In any case, this case is much more special than the general discussion, because arithmetic functions are distinguished by the property of multiplicativity: if gcd(m,n) == 1 then f(m*n) = f(m)*f(n). So in a sense, they are more deeply linked to the class of integers than, say, the binomial or factorial functions.

[kedlaya]

comment:8

Given that this discussion has gone dormant, I propose to mark this ticket as wontfix and close it.