# [seqfan] n consecutive perfect powers sum to a perfect power

jnthn stdhr jstdhr at gmail.com
Wed Dec 13 00:45:44 CET 2023

```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
```