A323614 -id:A323614 - cp's OEIS frontend

A323629 List of 6-powerful numbers (for the definition of k-powerful see A323395).

96, 128, 144, 160, 176, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400, 408, 416, 424, 432, 440, 448, 456, 464, 472, 480, 488, 496, 504, 512, 520, 528, 536

Offset: 1

Stan Wagon, Jan 20 2019

The set consists of 96, 128, 144, 160, 176, and all multiples of 8 that are greater than or equal to 192. The values 200, 216, 232, 248, 264, 280 are by Golan, Pratt, and Wagon; these are sufficient to give all further entries that are 8 (mod 16). Freiman and Litsyn proved that there is some M so that the list beyond M consists of all multiples of 8.

The linked file gives sets proving that all the given values are 6-powerful.

			a(1) = 96 because {1, 2, 7, 10, 11, 12, 13, 14, 16, 17, 21, 22, 27, 28, 32, 33, 35, 36, 37, 38, 39, 42, 47, 48, 51, 52, 53, 54, 56, 57, 63, 66, 67, 68, 71, 72, 73, 74, 77, 78, 79, 82, 88, 89, 91, 92, 93, 94} has the property that the sum of the i-th powers of this set equals the same for its complement in {1, 2, ..., 96}, for each i = 0, 1, 2, 3, 4, 5, 6.

S. Golan, R. Pratt, S. Wagon, Equipowerful numbers, to appear.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
G. Freiman and S. Litsyn, Asymptotically exact bounds on the size of high-order spectral-null codes, IEE Trans. Inform. Theory 45:6 (1999) 1798-1807.
Stan Wagon, Witnessing sets for the 6-powerful numbers
Stan Wagon, Overview table
Index entries for linear recurrences with constant coefficients, signature (2, -1).

Cf. A323614, A323610, A323395.

LinearRecurrence[{2,-1},{96,128,144,160,176,192,200},50] (* Harvey P. Dale, Aug 27 2025 *)

G.f.: -8*x*(x^6+2*x^2+8*x-12)/(x-1)^2. - Alois P. Heinz, Jan 25 2019

More terms added by Stan Wagon, Jan 25 2019

A306248 Smallest m for which 2n is not m-powerful (for the definition of k-powerful see A323395).

Original entry on oeis.org

1, 2, 1, 3, 1, 3, 1, 4, 1, 3, 1, 4, 1, 3, 1, 5, 1, 3, 1, 5, 1, 3, 1, 6, 1, 3, 1, 5, 1, 3, 1, 6, 1, 3, 1, 6, 1, 3, 1, 6, 1, 3, 1, 6, 1, 3, 1, 7, 1, 3, 1, 6, 1, 3, 1, 7, 1, 3, 1, 6, 1, 3, 1, 7, 1, 3, 1, 6, 1, 3, 1, 8, 1, 3, 1, 6, 1, 3, 1, 7, 1, 3, 1, 6, 1, 3, 1, 7, 1, 3, 1, 6, 1, 3, 1

Offset: 1

Stan Wagon, Jan 31 2019

This function is known as m*(2n). For odd n all values of m*(n) are 0.

			The bipartition {1,4}, {2,3} of {1,2,3,4} has equal first power-sums. But there is no such bipartition with equal power-sums for exponents 0, 1, and 2. Therefore a(2) = 2.

S. Golan, Equal moments division of a set, Math. Comp. 77 (2008) 1695-1712.
Stan Wagon, Data for n up to 128, updated Sep 29 2019.

Cf. A323395, A323610, A323629, A323614.

a(56) corrected by Stan Wagon, May 06 2019

a(72) corrected by Stan Wagon, May 24 2019

cp's OEIS Frontend

A323629 List of 6-powerful numbers (for the definition of k-powerful see A323395).

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