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.

2, 4, 8, 16, 32, 48, 96, 144, 192

Offset: 0

Author

Stan Wagon, Jan 13 2019

Keywords

• nonn
• hard
• more

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.

Examples

			{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.
		

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

Crossrefs

This sequence agrees with A222193 up to n=7.

Extensions

Edited by N. J. A. Sloane, Jan 19 2019

a(8) from Stan Wagon, Feb 04 2019