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%I #23 Jan 27 2022 20:57:41
%S 1,625,157464
%N a(n) is the smallest n-th power which can be represented as the sum of n distinct positive n-th powers in exactly n ways, or -1 if none exists.
%C From _Jon E. Schoenfield_, Dec 30 2021: (Start)
%C 222000^4 < a(4) < 4891341^4 = lcm(2829, 12259, 16359, 30381)^4 (see A039664, including the Wroblewski link).
%C 10000^5 <= a(5) < 12528^5 = lcm(72, 1044, 1392, 2088, 3132)^5 (see A063923, including the Waldby link; note that, although the terms of A063923 include 72, 144, 1044, 1392, and 2088, whose LCM is only 4176, the primitive solution in which the sum of 5 distinct 5th powers is 144^5 is 0^5 + 27^5 + 84^5 + 110^5 + 133^5 = 144^5, which is not the sum of 5 positive n-th powers).
%C Conjecture: a(6) = -1. (End)
%e For n = 2: 625 = 25^2 = 7^2 + 24^2 = 15^2 + 20^2.
%e For n = 3: 157464 = 54^3 = 6^3 + 36^3 + 48^3 = 12^3 + 19^3 + 53^3 = 27^3 + 36^3 + 45^3.
%Y Cf. A007666, A025303, A025401, A030052, A039664, A063923, A146756, A230718, A331675.
%K nonn,bref,hard,more
%O 1,2
%A _Ilya Gutkovskiy_, Dec 30 2021
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