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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.
6
2, 4, 8, 16, 32, 48, 96, 144, 192
OFFSET
0,1
COMMENTS
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.
LINKS
Joe Buhler, Shahar Golan, Rob Pratt, and Stan Wagon, Symmetric Littlewood polynomials, spectral-null codes, and equipowerful partitions, Mathematics of Computation, 329 (May 2021) 1435-1453; arXiv version, arXiv:1912.03491 [math.NT], 2019.
S. Golan, Equal moments division of a set, Math. Comp. 77 (2008) 1695-1712.
Stan Wagon, Overview table
EXAMPLE
{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.
CROSSREFS
This sequence agrees with A222193 up to n=7.
Sequence in context: A316750 A008382 A208742 * A326079 A222193 A217833
KEYWORD
nonn,hard,more
AUTHOR
Stan Wagon, Jan 13 2019
EXTENSIONS
Edited by N. J. A. Sloane, Jan 19 2019
a(8) from Stan Wagon, Feb 04 2019
STATUS
approved