Smallest k whose divisors sum to a multiple of n

[Allan Wechsler]

This is about the fourth time I have tried to compose this message, and I don't know if I can state the idea coherently.

Define s(n) = A000203(n) = the sum of the divisors of n.

Suppose we pick a number, n, and step down the sequence looking for occurrences of n. (I apologize for changing the meaning of n in midstream. This is an awkward consequence of trying to follow OEIS conventions.) Sometimes we won't find any. So let's relax the condition, and step down A000203 looking for multiples of n. By a straightforward application of Dirichlet's theorem, this process is guaranteed to succeed -- no matter what n is, there are multiples of n in A000203. 

This process produces several sequence ideas. For example, if I want to know how soon the first "hit" appears, the resulting sequences is A070982, "Smallest k such that n divides sigma(k)".

If I want to know the smallest possible multiple of n that occurs, the sequence I want is A283495, "Smallest k such that there is a number whose divisors sum to k*n". (Note that this isn't literally the smallest multiple of n, since the sequence gives k, not kn. I agree with that choice, I just wanted to say it out loud to avoid confusion.)

These two sequences fill two entries in a two-by-two table of possibilities.

In all cases we bop down A000203 looking for multiples of n.

One parameter is the stopping condition: we stop either when we find the first multiple of n, or the smallest multiple of n.

The other parameter is what we return as the value of A(n): we have found some sigma(k) = mn, and we can record either k (where did we find the multiple?) or m (which multiple did we find?). (Again, I apologize for the inconsistent choice of variable names. If you look at the sequence names, you'll see why I couldn't win.)

A070982 returns k for the first multiple.

A283495 returns m for the smallest multiple.

I am pretty sure that there is no sequence in OEIS that returns m for the first multiple. I am less sure, but suspect, that there is no sequence in OEIS that returns k for the smallest multiple.

I will write code to generate lots of terms of these two possibly-missing sequences, and propose the sequences themselves, sometime, when I get around to it, unless one of you good people beats me to it. These are probably one-liners in Pari or Sage, but I'm lame and don't know Pari or Sage, so I usually write Python code. But I will be happy if one of you bangs out some data first, and will happily cede authorship of the resulting sequences.

Note: It is often the case that the first multiple of n in A000203 is also the smallest multiple of n. You might in fact wonder if this is always the case. I am happy to provide the counterexample of n = 62. The first multiple of 62 in A000203 is sigma(48) = 124. But the smallest multiple of 62 in A000203 is sigma(61) = 62. I haven't checked assiduously to see if that's the smallest example where the earliest and smallest multiples differ: it was the one that jumped out at me when I was staring at the table for A000203. A list of the numbers whose earliest and smallest multiples in A000203 are different would be yet another interesting sequence, also probably not yet in OEIS.

Thanks for your patience in reading this blither.

-- Allan