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

1,1

Comments

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.

References

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

Links

Table of n, a(n) for n=1..49.

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).

Formula

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

Example

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.

Crossrefs

Cf. A323614, A323610, A323395.

Sequence in context: A153484 A060660 A258748 * A146992 A261287 A252689

Adjacent sequences: A323626 A323627 A323628 * A323630 A323631 A323632

Keyword

nonn,easy

Author

Stan Wagon, Jan 20 2019

Extensions

More terms added by Stan Wagon, Jan 25 2019

Status

approved