Curios about practical numbers that are square free.
Davide Rotondo
Practical numbers that are squarefree. up to n=100 and noticed that up to n=98 these numbers are k numbers such that k+1 or k-1 is a prime number. If neither k+1 nor k-1 is a prime number, then the difference between k and the immediately preceding prime number is a prime number, and the difference between the immediately following prime number greater than k and k is also a prime number. This holds true up to n=98, that is, 6942, since 6942 - 6917 = 25. Now I wonder if at least one of the differences is always a prime number. What do you think?
Amiram Eldar
The next terms in A265501 after 203082 with composite differences are 458094, 630762, 668382, 677274, 693462, 745446, 818862, 836022, 843018, 963534, 966702, 979602, 1072266, 1133106, 1189518, 1251354, ...
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Gareth McCaughan
Hello,
The least counterexample is n = 1630: A265501(n) = 203082, and its differences from the preceding prime and the next prime are both composites: 25 and 35, respectively.
The next terms in A265501 after 203082 with composite differences are 458094, 630762, 668382, 677274, 693462, 745446, 818862, 836022, 843018, 963534, 966702, 979602, 1072266, 1133106, 1189518, 1251354, ...
Best,Amiram
On Tue, Dec 2, 2025 at 12:49 PM Davide Rotondo <david...@gmail.com> wrote:
Hello, dear fans of mathematical sequences. I want to bring to your attention a curious thing I noticed today. I analyzed the sequence A265501
Practical numbers that are squarefree. up to n=100 and noticed that up to n=98 these numbers are k numbers such that k+1 or k-1 is a prime number. If neither k+1 nor k-1 is a prime number, then the difference between k and the immediately preceding prime number is a prime number, and the difference between the immediately following prime number greater than k and k is also a prime number. This holds true up to n=98, that is, 6942, since 6942 - 6917 = 25. Now I wonder if at least one of the differences is always a prime number. What do you think?
A few words of advice for Davide:
Any time you have a conjecture like this one that looks like "for all n for which [something unusual] is true, [something rather common] is also true", you should ask yourself: is that second not-so-unusual thing common enough that this might hold _just by chance_ for a lot of early cases?. And if it is, you should want quite a lot of evidence before expecting the conjecture to be true in general.
In this case, it looks as if "practical numbers" > 1 are always even and commonly have quite a lot of small factors. In this case, n +- 1 and n +- nearby primes will have fewer-than-usual small factors, and (in the latter case) tend to be small, which means they will often be prime.
Even if all we knew were that these numbers are even, of the 500
even numbers <= 1000 all but 35 either are prime+-1 or have
their difference to the nearest prime on one or other side being
prime.
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