Computing bounded solutions to linear Diophantine equations with the sum of divisors

[Max Alekseyev]

SeqFans,

I'm pleased to let you know about my new preprint on an efficient computational method for solving linear equations with sigma(n), that is,

a * sigma(n) = b*n + c

and a first public release of the corresponding SageMath software implementation.

Some of you may have seen many large solutions to such equations that I recently added to the OEIS. This preprint explains how they were computed.

Any comments are welcome!

Preprint: http://arxiv.org/abs/2601.17832
Software: https://github.com/maxale/multiplicative_functions

Regards,

Max

[Alex Violette]

Hi Max,
     I did want to say, this is an interesting read(and I like the idea of the name prime wheel).
I was going to email you about this sometime soon but I do want to point out that a sequence for sigma(n)-2n=54 already exists(A391617) and I have one more in the works(A391615) that I will inform about in a separate email.
About almost perfect numbers that are not powers of 2 and quasi-perfect numbers, I have investigated them and numbers where sigma(n)-2n is odd enough for me to say this: I am honestly not optimistic that these kinds of numbers exist though I cannot outright rule them out. I'm honestly surprised I have not seen much speculation that these kinds of numbers "don't exist" compared to odd perfect numbers.
     Also about the number 20055918935605248255: I have been aware of its property |(sigma(n)-2n)|=d(n)-2 and the number itself since sometime in late October 2023 but I was not aware of the particular OEIS sequence(A066229) it is part of or (x-1) perfect numbers in general prior to its inclusion in the OEIS. That said I don't expect any more odd terms to be found for that sequence anytime soon most likely though I will point out that I do have odd numbers that satisfy the following equations for the following:
sigma(n)-2n=d(n)/4-2:  22835519561796362321806710606606494356248711213668106239745
|sigma(n)-2n=d(n)/2-2|: 327985795693319489015162672668999935, 605700324781758737374486971458156409352525408736624639745

|sigma(n)-2n|=d(n)*4-2|: 805983436545, 1057975556279283743745, 1555427957767853025171344507139206163402754115820061695

Turns out, I do have an odd solution for A066230(98354828344320469458616065) which I have overlooked until now which I have just added to the OEIS.

Best, 
Alex Violette

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