L-perfect numbers

[Žiga Pirc]

Hello, 

in A066218 we find a way to generilize perfect numbers (along with abundant, deficiend, amicable, semiperfect, ect. numbers), proposed by Joseph L. Pe.

A f-perfect number n is such a number that 2*f(n) = sum(d|n) {f(d)}. By this definition "normal" perfect numbers are id-perfect, since their corresponding function is the identity. A066218 are the sigma-perfect numbers and if you look at sequences refrencing A066218 you find about 20 sequences with different functions, for example A066367, the "floor(prime(n)/n)"-perfect numbers. There are likely only 5 of them, I checked up to 10^8 and found none above 169.

L-perfect or linear-perfect numbers are such numbers where f(n) = m*n+a. A table (would this be worth a oeis entery?) of smallest number for a given linear function is:

1, 2, 2, 2, 2, 2
6, 4, 14, 15, 22, 9
6, 8, 4, 52, 14, 68
6, 16, 110, 4, 130, 33714176
6, 32, 8, 592, 4, 656
6, 64, 884, 315, 1012, 4
Where top left f(n) = 0, top right f(n) = 5 and bottom left f(n) = 5*n. f(n) = 0*n+a will always give 2, since such perfect numbers are primes, f(n) = m*n+0 will always give 6, since these are our normal perfect numbers. f(n) = m*n+1 give 2^(m+1), but further terms are not connected to powers of 2, for 2n+1 8 might even be it's only perfect number, if another exists it's > 10^8.

Subset of L-perfect numbers are where m = 1 and a changes. For a>0 the sequence of smallest "n+a"-perfect numbers is: 4, 14, 15, 22, 9, 124, 27, 38, 33, 46, ...
There exists at least 1 such number up to a = 250.
If we look at negative a this maybe isnt a case; smallest "n-60"-perfect number - if it exists - is bigger than 10^9. Or maybe this is a case similar to 3*n+5 from the table above.

You could ask if generalizing a not that well understood concept such as perfect numbers is even usefull. Well, "n+1"-perfect numbers A066229 have a connection to groups for example, so some other connections can be found.

Does someone has an insight into them or some interesting functions to generate f-perfect numbers(for example primepi-perfect numbers are not in oeis yet, 1, 6, 10, 63, 164, 188, 374, 418, 441, 602, ...)? Finding functions that generate same set of numbers and are not multiples of each other could be interesting, if such functions even exist.

Best,

Žiga 

[Max Alekseyev]

L-perfect numbers can be obtained with my approach/code. See Section 5.3 in https://arxiv.org/abs/2601.17832

Regards,

Max

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[Žiga Pirc]

I set up sage few days after your reply, works great!

Is there a way to only print out results and not "[]" when it finds nothing? Im searching for a 8n+3 perfect number and I searched up to 224403121196654400 so it's calling res_solve_sigma_abc 80640 times and it ouputs 80640 empty solutions. As I understand lambda x : [x] outputs it?

Žiga

torek, 21. julij 2026 ob 16:15:50 UTC+2 je oseba Max Alekseyev napisala:

[Max Alekseyev]

Hi Žiga,

As this kind of questions is too technical and not of general interest, it's better to email them directly to me (not to SeqFan maillist).

Please do so and provide an actual code showing how you call res_solve_sigma_abc().

f_proc argument is used for additional filtering of solutions (e.g. when there are too many extraneous ones to store them); the default value "lambda x: [x]" essentially means no filtering with all solutions passing through.

Regards,

Max

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