Empirical Relationship Between A248234 (Zeta(5)), A183861, and the Diff of Squares * Sum of Cubes Combinatory Algorithm

[Rui Ferreira]

Hello SeqFan,

I observed an interesting connection between the OEIS sequences A248234 and A183861. After running a combinatorial calculation based on a specific combinatorial formula based on differences of squares and sums of cubes of integer pairs (Python code at https://github.com/rf-iasys/OEIS/blob/main/OEIS_A248234_A183861.py), the results I obtained match A248234 almost exactly being the differences between the two corresponding exactly to A183861.

When you subtract the values of A183861 from A248234, you get the exact same numbers as the ones produced by my combinatorial calculation.

Index |     Computed | OEIS A248234 |   Diff | A183861 (Formula)
----------------------------------------------------------------
    1 |           27 |           27 |      0 | N/A                
    2 |          175 |          176 |     -1 |                1
    3 |          637 |          639 |     -2 |                2
    4 |         1701 |         1706 |     -5 |                5
    5 |         3751 |         3759 |     -8 |                8
    6 |         7267 |         7279 |    -12 |               12
    7 |        12825 |        12842 |    -17 |               17
...
 1998 | 63808227874027 | 63808229205359 | -1331332 |     1331332
 1999 | 63936035990001 | 63936037322666 | -1332665 |     1332665
 2000 | 64064036010001 | 64064037343999 | -1333998 |     1333998

This observation shows an interesting relationship between the sequences and highlights the potential for further investigation. While I don’t have a formal mathematical background, the results from the combinatorial calculations and the alignment with both A248234 and A183861 are compelling. I hope the OEIS community will find this connection worth exploring.

Just to complement that many other OEIS series surprisingly have a direct combinatory construction as can be observed at https://github.com/rf-iasys/OEIS

Again, I do not have a math background that allow me to independently understand the implications of such observations. Please take this in consideration in your more than welcome comments.

[Gareth McCaughan]

On 14/02/2026 22:00, Rui Ferreira wrote:
> Hello SeqFan,
>
> I observed an interesting connection between the OEIS sequences
> A248234 and A183861. After running a combinatorial calculation based
> on a specific combinatorial formula based on differences of squares
> and sums of cubes of integer pairs (Python code at
> https://github.com/rf-iasys/OEIS/blob/main/OEIS_A248234_A183861.py),
> the results I obtained match A248234 almost exactly being the
> differences between the two corresponding exactly to A183861.
>
> When you subtract the values of A183861 from A248234, you get the
> exact same numbers as the ones produced by my combinatorial calculation.

As with your previous calculation having this form, the business with x
and y=x(b-a) obfuscates what's really going on. Your compute_elements
function does the following:

for a<b with a+b <= n_end:
  x = (b^2-a^2)(b^3+a^3)
  y = x(b-a)
  if y=x then add this value to your list

which is exactly equivalent to the following:

for all a with 2a+1 <= n_end:
  b = a+1
  x = (b^2-a^2)(b^3+a^3)
  add this value to your list

(General note: If all your "combinatorial calculation" things are of the
form "for all a,b write x=... and y=x(a-b); then consider only the cases
where y=x" then they are all obfuscated versions of a simpler
calculation that just says "for all a write b=1 and then x=...".)

... which is exactly equivalent to the following:

for all a <= (n_end-1)/2:
  add (2a+1)(a^3+(a+1)^3) to your list

or:

for all a <= (n_end-1)/2:
  add (2a+1)^2 (a^2+a+1) to your list

So your observation is that it appears that

A248234(n) = floor(1/((n+1)^-5 + (n+2)^-5 + (n+3)^-5 + ...)

equals

(2n+1)^2 (n^2+n+1) + (n-2) + ceiling(((n-1)^2-1)/3)

for n >= 3. (For n=2 we need A183861(1) which is special-cased; for n=1,
A183861(0) isn't defined at all.)

This is in fact _already in A248234_, though not in the same form; it's
described as a "Conjecture" made by Kotesovec, but Sahu's paper linked
in the OEIS entry proves it (in, again, a slightly different form; there
are generally lots of different ways to write down things of the form
"nearest integer to polynomial in n with coefficients rational with
small denominators"; I have to say I really don't like Kotesovec's
version, which makes things look distinctly more complicated than they
really are).

There will be similar rounded-polynomial expressions for the remainder
terms in other zeta(n); the case n=3 is in A248230 (and, again, in
Sahu's paper).

Your observation amounts to writing the relevant integerified polynomial
as a combination of one particular quartic polynomial that has the right
n^4 and n^3 terms, and a quadratic correction that happens to be in
OEIS. I don't think there's any significance to the particular quartic
polynomial you've chosen, beyond the fact that its highest two
coefficients are the ones you need.

--
g