New Fini Full sequence

[Martin Musatov]

Hello, I was just looking for feedback, comments, observations, on this potential new series:

Thanks,

Martin

Prime numbers that can be written as half the sum of three squares in exactly one unique way.

2, 3, 5, 7, 11, 23, 29, 71

There are only eight terms and no prime p > 71 has a unique solution due to class number growth restrictions under the Stark-Heegner theorem.

{ prime p : 2*p in { A094942 } }.

2: (2^2 + 0^2 + 0^2) / 2 = (4 + 0 + 0) / 2 = 2

3: (2^2 + 1^2 + 1^2) / 2 = (4 + 1 + 1) / 2 = 3

5: (3^2 + 1^2 + 0^2) / 2 = (9 + 1 + 0) / 2 = 5

7: (3^2 + 2^2 + 1^2) / 2 = (9 + 4 + 1) / 2 = 7

11: (3^2 + 3^2 + 2^2) / 2 = (9 + 9 + 4) / 2 = 11

23: (6^2 + 3^2 + 1^2) / 2 = (36 + 9 + 1) / 2 = 23

29: (7^2 + 3^2 + 0^2) / 2 = (49 + 9 + 0) / 2 = 29

71: (9^2 + 6^2 + 5^2) / 2 = (81 + 36 + 25) / 2 = 71

[T. S.]

The numbers are one thing, but OEIS is also a place where such information and related detail will be found: "There are only eight terms and no prime p > 71 has a unique solution due to class number growth restrictions under the Stark-Heegner theorem." This is why I like such cases.

[Sean A. Irvine]

Hi,

My opinion is that this sequence is too short with small terms and not important enough to justify its own sequence in the OEIS.

A comment could instead be added to A094942.

Sean.

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[William Keith]

But if one enters these terms into the OEIS, A094942 is not a
suggested match under any transformation or the like. Yes, obviously
the numbers are "just" small primes, but the fact that this condition
yields a finite sequence is non-obvious and is the interesting thing
about it. I would say yes, it is a reasonable inclusion.

Best,
William Keith

[Elijah Beregovsky]

What about the sequence families generated if you swap the numbers in the definition? Like “Prime numbers that can be written as half the sum of three squares in exactly n unique ways”, “ Prime numbers that can be written as half the sum of n squares in exactly one unique way” or maybe even “Prime numbers that can be written as one nth the sum of three squares in exactly one unique way”. Is there anything one might easily tell about any of those? Does any one of the families produce finite sequences only? Because if that is the case, you may be able to add the family to the OEIS as an irregular triangle. Just my two cents :)

Elijah