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A323395 a(n) is the smallest n-powerful number, that is, the smallest positive integer such that {1,2,...,a(n)} admits a partition into A and B so that the sum of the j-th powers of A equals the sum of the j-th powers of B, for all j = 0, 1, ..., n.

%I A323395 #59 Jun 17 2022 02:16:33
%S A323395 2,4,8,16,32,48,96,144,192
%N A323395 a(n) is the smallest n-powerful number, that is, the smallest positive integer such that {1,2,...,a(n)} admits a partition into A and B so that the sum of the j-th powers of A equals the sum of the j-th powers of B, for all j = 0, 1, ..., n.
%C A323395 The determination of these values is difficult. Early work is due to D. Boyd. The Golan paper cited has references to the earlier work. Work of Buhler, Golan, Pratt, and Wagon (2021) showed that a(8) is 192.
%H A323395 Joe Buhler, Shahar Golan, Rob Pratt, and Stan Wagon, <a href="https://doi.org/10.1090/mcom/3612">Symmetric Littlewood polynomials, spectral-null codes, and equipowerful partitions</a>, Mathematics of Computation, 329 (May 2021) 1435-1453; <a href="https://arxiv.org/abs/1912.03491">arXiv version</a>, arXiv:1912.03491 [math.NT], 2019.
%H A323395 S. Golan, <a href="https://doi.org/10.1090/S0025-5718-08-02072-3">Equal moments division of a set</a>, Math. Comp. 77 (2008) 1695-1712.
%H A323395 Stan Wagon, <a href="/A323395/a323395.pdf">Overview table</a>
%e A323395 {1, 2, 7, 10, 11, 12, 13, 14, 16, 17, 21, 22, 27, 28, 32, 33, 35, 36, 37, 38, 39, 42, 47, 48} and its complement {3, 4, 5, 6, 8, 9, ..., 43, 44, 45, 46} in {1, 2, ..., 48} have equal power-sums for exponents 0 to 5, the key step in showing that a(5) = 48.
%Y A323395 This sequence agrees with A222193 up to n=7.
%K A323395 nonn,hard,more
%O A323395 0,1
%A A323395 _Stan Wagon_, Jan 13 2019
%E A323395 Edited by _N. J. A. Sloane_, Jan 19 2019
%E A323395 a(8) from _Stan Wagon_, Feb 04 2019