Second opinion for A186655(10)?

[Martin Fuller]

Could a seqfan give a second opinion about A186655(10) and A186657(10)? I asked the original author and he is happy for someone else to check.

OEIS has A186655(10)=2070553183, A186657(10)=1625380302.
I get A186655(10)=2070553182, A186657(10)=1625380303.

I calculated A186655(10) in two different ways and get the same result.

Martin Fuller

PS For n=11 I get the same results as OEIS, which means there is an off-by-one difference for the related sequences A186656 and A186658 for n=10 and 11. All the other related sequences - A186649 etc - I get identical up to n=11.

[David desJardins]

Perhaps the off-by-1 is due to ambiguity in the definition?

If A186655 is defined to be, "Total number of positive integers below 10^n requiring 5 positive biquadrates in their representation as sum of biquadrates," then does it include 10^n itself, or not?

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[Kevin Ryde]

David desJardins <da...@desjardins.org> writes:
>
> does it include 10^n itself, or not?

I tried a program which claims 10^10 itself needs 5 biquadrates,
so potentially. 10^6 also needing 5 seems ok (not in 5s A186655).

But yeah 5s 1 too big and 6s 1 too small looks more like somebody
in the wrong count.

[Martin Fuller]

I'm confident 10^n is meant to be excluded:

* A186655(2)=7, whereas it would be 8 if it included A002377(100)=5.

* A186656 is described as "n-digit numbers requiring 5 positive biquadrates" and the formula is "a(n) = A186655(n) - A186655(n-1)", which only makes sense if 10^n is excluded from A186655.

[M F Hasler]

On Sun, Jan 4, 2026 at 5:34 PM 'Martin Fuller' via SeqFan <seq...@googlegroups.com> wrote:

Could a seqfan give a second opinion about A186655(10) and A186657(10)?

My opinion is that the sequence's definition and description is incomplete.

To the least, the name ("...requiring 5 positive...') should specify "exactly" vs "at least", 

and various other "details" that could be guessed but should be stated or at least illustrated in some interesting examples,

in particular for cases that might be ambiguous.

(Although "below" does mean "strictly less than", that could be illustrated,

[cf. your 2nd email "I'm confident 10^n is meant to be excluded..."],

maybe also whether the biquadrates have to be different or not, etc.

(Some people could say that 1 + 1 + 1 + 1 + 1 contains only one biquadrate, namely 1, 

because there is no other one in the sum. Of course we would all guess that they are counted 

with multiplicity, otherwise it would say "...distinct...", but it still could be said explicitly or illustrated through an example.)

I think it is always a good idea to clarify all such potential ambiguities 

(with proposal of a corresponding edit of the description of the sequence)

before starting calculations.

Also, there's a cryptic huge formula but not some simpler formulas like

A186655(n) = # { k < 10^n |  Axxx(k) = 5 }

where Axxx(k) gives the number of positive biquadrates required to sum up to k.

(which sequence number is that? missing only in XREFs or in OEIS ? Shouldn't one start there, then?) 

- Maximilian

[David desJardins]

On Tue, Jan 6, 2026 at 5:09 AM M F Hasler <mha...@dsi972.fr> wrote:

Also, there's a cryptic huge formula but not some simpler formulas like

A186655(n) = # { k < 10^n |  Axxx(k) = 5 }

where Axxx(k) gives the number of positive biquadrates required to sum up to k.

(which sequence number is that? missing only in XREFs or in OEIS ? Shouldn't one start there, then?) 

https://oeis.org/A002377