[seqfan] Re: n consecutive perfect powers sum to a perfect power

[Jack Brennen]

I've been looking for a solution for 9 consecutive powers.

If one exists, it's greater than 6.8 * 10^19.

Note that any such solution must include an odd power (cube, 5th power, 
etc.).

This is because the sum of 9 consecutive squares is of the form 9k^2 + 
60, which can't be a perfect power of course, because it's divisible by 
3 with multiplicity of 1.



On 12/13/2023 8:21 PM, Tim Peters wrote:
> I'll attach a Python program that works to cut off sums when it becomes
> futile to pursue them. I assume your algorithm isn't doing that, because
> this only takes about 2 seconds to finish 10^10 (running under PyPy).
>
> At 10^14 it took noticeable (very noticeable ;-) ) time, but still didn't
> find a solution for sums of 9 or 10 consecutive powers. It found 744
> solutions in all, the longest being a sum of 59924 consecutive powers,
> starting at 37711881, summing to 92121066512784 (the square of 9597972).
>
> The code:
>
> from math import isqrt
>
> TOP = 10**10
> SHOW_DELETES = False
>
> ppset = {1}
> for i in range(2, isqrt(TOP) + 1):
>     power = i
>     while (power := power * i) <= TOP:
>         ppset.add(power)
> pp = sorted(ppset)
> npp = len(pp)
> MAXPP = pp[-1]
> print(f"len(pp)={npp:_} through {TOP=:_}; {MAXPP=:_}")
>
> sums = pp[:]
> nterms = 0
> delta = 1
> while sums:
> ##    # Crucial invariant, but expensive to check.
> ##    assert all(sums[i] == sum(pp[i : i + delta])
> ##               for i in range(len(sums)))
>      winner = -1
>      for i, s in enumerate(sums):
>          if s in ppset:
>              winner = s
>              assert sum(pp[i : i + delta]) == s
>              break
>      if winner > 0:
>          nterms += 1
>          print(nterms, delta, winner,
>                f"starting at pp[{i}] = {pp[i]}, {len(sums)=:_}")
>      if len(sums) + delta > npp:
>          sums.pop()
>          assert len(sums) + delta == npp
>      for i in range(len(sums)):
>          sums[i] += pp[i + delta]
>          if sums[i] > MAXPP:
>              if SHOW_DELETES:
>                  print("    deleting", len(sums) - i,
>                        "of", len(sums),
>                        "at delta", delta)
>              del sums[i:]
>              break
> ##    assert sums == sorted(set(sums))
>      delta += 1
>
> On Wed, Dec 13, 2023 at 12:44 AM jnthn stdhr <jstdhr at gmail.com> wrote:
>
>> Howdy, all.
>>
>>    What is the least perfect power m (in A001597) that is the sum of n
>> consecutive perfect powers.
>>
>>    The sequence isn't in the database, and begins 1, 25, 441, 100, 169, 289,
>> 121, 2395417249, -1, -1, 676, 232324, -1, -1, -1, 64866916, 3721,
>> 3622354596, 279936, ..., with -1 representing no solution found up to
>> ~10^10.
>>
>>    For the first few terms, we have:
>>
>> {1}=1, {9+16}=25, {128+144+169}=441, {16+25+27+32}=100, etc.
>>
>>    Should I add this? If so, up to a(8) only, or include the -1s?
>>
>>   When I first searched for solutions, the maximum value of the set of
>> perfect powers was ~10^8, and both a(8) and a(18) came out -1.  But when I
>> increased the max to 10^10 solutions for those two terms were found.  At
>> 10^8 I was able to get to 100+ terms in a somewhat reasonable time, with
>> solutions becoming more and more sparse.  At 10^10, things get very bogged
>> down, but more solutions are found along the way.  Also, the erratic nature
>> of the terms seems to persist.
>>
>>    Can a solution always be found if the set of perfect powers is large
>> enough?
>>
>> -Jonathan
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
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