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%I A323629 #33 Jul 01 2023 16:21:19
%S A323629 96,128,144,160,176,192,200,208,216,224,232,240,248,256,264,272,280,
%T A323629 288,296,304,312,320,328,336,344,352,360,368,376,384,392,400,408,416,
%U A323629 424,432,440,448,456,464,472,480,488,496,504,512,520,528,536
%N A323629 List of 6-powerful numbers (for the definition of k-powerful see A323395).
%C A323629 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.
%C A323629 The linked file gives sets proving that all the given values are 6-powerful.
%D A323629 S. Golan, R. Pratt, S. Wagon, Equipowerful numbers, to appear.
%H A323629 G. Freiman and S. Litsyn, <a href="https://doi.org/10.1109/18.782100">Asymptotically exact bounds on the size of high-order spectral-null codes</a>, IEE Trans. Inform. Theory 45:6 (1999) 1798-1807.
%H A323629 Stan Wagon, <a href="/A323629/a323629.txt">Witnessing sets for the 6-powerful numbers</a>
%H A323629 Stan Wagon, <a href="/A323610/a323610.pdf">Overview table</a>
%H A323629 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1).
%F A323629 G.f.: -8*x*(x^6+2*x^2+8*x-12)/(x-1)^2. - _Alois P. Heinz_, Jan 25 2019
%e A323629 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.
%Y A323629 Cf. A323614, A323610, A323395.
%K A323629 nonn,easy
%O A323629 1,1
%A A323629 _Stan Wagon_, Jan 20 2019
%E A323629 More terms added by _Stan Wagon_, Jan 25 2019

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