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

[hv at crypt.org]

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.

So I thought, maybe we should look instead at:
  a(n) := min(k: lucky(k - 2^i) forall i in {1..n})

I expected this would grow reasonably smoothly, but enough faster than 2^n
to give supporting evidence that further elements of A057652 are unlikely.
Instead, I found that the existing terms of A057652 yielded the sequence
[ 1, 3, 5, 11, 17, 647, 647, 647, 647, 647 ], and I didn't find the next.

Letting f(k) represent the highest power of 2 achieved for a given k, I found
the following for 0 < k < 1e6, f(k) >= 5:

f(k) = 5: 2411 15263 67931 104555 152099 167681 226901 245843 248993 338237 
  354581 403595 434843 567353 594359 627497 645299 701495 762317 880919 
  923597 945983
f(k) = 6: 7199 16397 66581 157175 303839 328079 348617 350927 513503 628085 
  932669
f(k) = 7: 6479 290531 323585 472601
f(k) = 8: 772433
f(k) = 9: 647

I guess this is pretty close to what you'd expect, though I hadn't thought
it would be quite so lumpy. I guess a(10) should appear somewhere between
1e6 and 3e6. It does suggest though that f(647)=9 was super unlikely.

Given my perl code managed up to 1e6 easily enough, I think rewriting in C
would let me generate a list of the lucky numbers up to 1e8 fairly easily,
maybe closer to 1e9. Let me know if that would be useful.

Hugo