%I #26 Jul 24 2020 04:23:56

%S 2,8,11,44,27,33,42,83,51,62,72,83

%N Largest m for which m^n = Sum_{x in S} x^n has no solution S subset of {1, ..., m-1}.

%C Row n of table A332065 lists all s for which there is some S subset of {1,...,m-1} with s^n = Sum_{x in S} x^n. This is the case for all sufficiently large s (cf. reference there). Here we give the largest integer not in this list.

%C Sequence A030052 lists the smallest m for which there is a solution, so a(n) >= A030052(n) - 1. We have a(9) = 51 = A030052(9) + 4, a(10) = 62 = A030052(10) - 1, a(11) = 72 = A030052(11) + 4. - _M. F. Hasler_, May 14 2020, edited Jul 19 2020

%F a(n) = A030052(n) - 1 or a(n) > A030052(n).

%F a(n) < A001661(n)^(1/n).

%e For n=1, we have m^n = (m-1)^n + 1^n, so S = {1, m-1} is a solution for all m > 2, but 2^n > 1^n and therefore no solution with m = 2 = a(1).

%e For n=2, we have a solution to m^n = Sum_{x in S} x^n for S subset of {1,...,m-1} for all m > 8 (cf. FORMULA in A332065), but no solution with m = 8 = a(2).

%Y Cf. A332065, A030052, A332066.

%K nonn,hard,more

%O 1,1

%A _M. F. Hasler_, Apr 19 2020

%E a(8) - a(12) from _M. F. Hasler_, Jul 23 2020