[seqfan] Re: A057652 has no more terms below 10^4

[Alonso Del Arte]

1/743rd of infinity is still an infinity, but at least to me it feels
somehow more manageable. There is value to testing those largish small
numbers that are within the reach of our tools even if no proof suggests
itself in the process. In this particular case, I would wager that the
proof, if it can be found, employs modular arithmetic in some way.

David Wilson's write-up for the Sequence of the Day today shows an
encouraging example where number crunching refutes a very obvious and
reasonable assumption.

Al

P.S. If you're curious how the Sequence of the Day looks in transclusion,
you can see it at <http://oeis.org/wiki/User:Alonso_del_Arte/Test_Area>.

On Sun, Oct 17, 2010 at 5:29 AM, <hv at crypt.org> wrote:
>
>
> A negative result:
>
> I tried constructing the minimal modular pattern after each new lucky
> number was sieved out, and for mod m, checked how many values mod m were
> candidates for this sequence taking into account powers up to the first
> 2^k > m expecting that they would fairly rapidly descend to zero, thus
> proving that the sequence is finite.
>
> Instead (assuming my code is correct) I get this:
>
> After sieving 2, lucky pattern now has 1 value mod 2
> Up to 2^1, 1 value mod 2 is possible
> Up to 2^2, 1 value mod 2 is possible
> After sieving 3, lucky pattern has 2 values mod 6
> Up to 2^2, 1 value mod 6 is possible
> Up to 2^3, 1 value mod 6 is possible
> After sieving 7, lucky pattern has 12 values mod 42
> Up to 2^3, 4 values mod 42 are possible
> Up to 2^4, 3 values mod 42 are possible
> Up to 2^5, 2 values mod 42 are possible
> Up to 2^6, 2 values mod 42 are possible
> After sieving 9, lucky pattern has 32 values mod 126
> Up to 2^6, 3 values mod 126 are possible
> Up to 2^7, 3 values mod 126 are possible
> After sieving 13, lucky pattern has 384 values mod 1638
> Up to 2^7, 18 values mod 1638 are possible
> Up to 2^8, 18 values mod 1638 are possible
> Up to 2^9, 18 values mod 1638 are possible
> Up to 2^10, 18 values mod 1638 are possible
> Up to 2^11, 18 values mod 1638 are possible
> After sieving 15, lucky pattern has 1792 values mod 8190
> Up to 2^11, 57 values mod 8190 are possible
> Up to 2^12, 57 values mod 8190 are possible
> Up to 2^13, 57 values mod 8190 are possible
> After sieving 21, lucky pattern has 5120 values mod 24570
> Up to 2^13, 132 values mod 24570 are possible
> Up to 2^14, 130 values mod 24570 are possible
> Up to 2^15, 124 values mod 24570 are possible
> After sieving 25, lucky pattern has 24576 values mod 122850
> Up to 2^15, 405 values mod 122850 are possible
> Up to 2^16, 354 values mod 122850 are possible
> Up to 2^17, 352 values mod 122850 are possible
> After sieving 31, lucky pattern has 737280 values mod 3808350
> Up to 2^17, 5678 values mod 3808350 are possible
> Up to 2^18, 5533 values mod 3808350 are possible
> Up to 2^19, 5265 values mod 3808350 are possible
> Up to 2^20, 5247 values mod 3808350 are possible
> Up to 2^21, 5183 values mod 3808350 are possible
> Up to 2^22, 5123 values mod 3808350 are possible
> [After sieving 33, lucky pattern would have 7864320 values mod 41891850]
>
> So above 2^22, you need only check about one number in 743 as a possible
> candidate for A057652; however, this approach does not look likely to yield
> a proof that the sequence is finite (and maybe complete).
>
> Hugo
>
>
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