Three‐way tie among at least three of the final digits of prime numbers.

[Davide Rotondo]

Here the 56 prime numbers ≤ 100 000 for which, upon their entry into the “race” of last digits {1, 3, 7, 9}, a three‐way tie among at least three of these digits occurs for the first time. In increasing order:

11, 17, 19, 23, 31, 37, 41, 47, 59, 67,
97, 109, 113, 149, 151, 157, 191, 211, 241, 257,
269, 277, 337, 431, 439, 443, 461, 463, 499, 547,
571, 821, 1481, 1567, 1597, 1609, 1613, 1627, 3449, 3719,
3727, 3761, 3767, 4271, 10 177, 10 529, 10 567, 10 601, 10 859, 10 889,
10 937, 56 269, 56 333, 56 431, 56 437, 56 711.

What do you think dear seqfans? Is this sequence infinite?

Is this interesting to be insert in the OEIS?

See you soon

Davide

[Jon Wild]

I don't know what the real sequence is (it looks like the next numbers
after 11 might be 31, 41, 61...), but if you think it's worth submitting
for consideration, then I would say that it's worth you actually working
out the correct beginning of the sequence, instead of using a chatbot to
guess at the numbers!!

Also your definition says a number appears when a three-way tie is found
"for the first time", which clearly needs to be removed for the sequence
to keep growing.

Jon


On 2025-10-20 11:15 AM, Davide Rotondo wrote:
> You are right, sorry I used MathGPT. What’s the real sequence?
>
> Il giorno lun 20 ott 2025 alle 5:07 PM Jon Wild <jon....@mcgill.ca
> <mailto:jon....@mcgill.ca>> ha scritto:
>
> Hi Davide, I don't understand. At the entrance of 17, we have the
> following counts:
>
> final digit 1:  11      (1)
> final digit 3:  3,13    (2)
> final digit 7:  7,17    (2)
> final digit 9:          (0)
>
> There are respectively counts of 1, 2, 2, and 0. Where is the three-
> way tie?
>
> Thanks,
>
> Jon

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[Allan Wechsler]

Three-way ties under 100 appear at 11, 31, 41, 71, and 89, according to my quick hand-calculation. There is no tie at 61 unless I blundered.

I personally don't find this sequence particularly compelling, but that's just my own esthetic sense. Why are we interested in remainders modulo 10, rather than any other number? Why threeway ties and not fourway? (I haven't found any four way ties yet.)

Probabilistically, this process resembles a random walk on a three dimensional tetrahedral grid divided into six "sextants", and looking for places where we cross from one sextant to another. So I would expect the sequence to be infinite but to get sparse fairly quickly, like most similar "codimension 1" random walk events. (I guess this answers the "why not fourway?" question: for that, the events are of codimension 2, and we would expect only a finite number of them.)

The most similar sequence I have been able to find in OEIS is A098044, which deals with primes of type 3n+1 and 3n+2. I can't find an entry for 4n+1 versus 4n+3, because after starting 5, 17, 41, the 4n+3 class opens up a commanding lead and I can't guess when 4n+1 is likely to close the gap.

-- Allan

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[Alex Violette]

Hey Allen, I think this is probably what you are looking for on the OEIS for the 4n+1 vs 4n+3 case:  A007351 - OEIS
I've known about how this race goes for years now(Ex: Infinitely many cases where 4n+1 surpassed 4n+3)

Regards,
Alex Violette

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[Allan Wechsler]

Thanks, Alax, you nailed it.

-- Allan

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