The sets below witness that the given integers are 6-powerful. Knowing the values below is sufficient to get the entire infinite set because if m and n are in 6-powerful, so is m+n.

96
{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}

128
{1,4,6,7,10,11,13,16,18,19,21,24,25,28,30,31,34,35,37,40,
41,44,46,47,49,52,54,55,58,59,61,64,66,67,69,72,73,76,78,
79,81,84,86,87,90,91,93,96,97,100,102,103,106,107,109,112,
114,115,117,120,121,124,126,127}

144
{1,2,8,9,10,11,12,15,18,20,21,22,26,27,30,31,35,36,37,38,
42,43,46,47,51,52,53,55,58,61,62,63,64,65,71,72,75,76,77,
78,79,85,86,88,89,91,95,96,97,100,101,104,105,106,111,
112,113,116,117,120,121,122,126,128,129,131,132,138,139,
140,141,142}

160
{1,2,8,9,10,11,13,14,17,20,22,23,25,26,29,34,35,36,37,38,
43,44,45,46,47,52,55,56,58,59,61,64,67,68,70,71,72,73,79,
80,83,84,85,86,87,92,95,96,98,99,101,104,107,108,110,111,
112,113,119,120,121,122,128,129,130,131,133,134,137,140,
142,143,145,146,149,154,155,156,157,158}

176
{1,4,5,6,10,13,14,16,19,20,22,23,25,26,28,30,34,35,38,40,
42,43,46,47,49,51,54,55,59,61,63,64,66,67,69,70,73,75,76,
79,83,84,85,88,90,91,95,96,97,99,100,103,105,106,109,112,
115,117,119,120,121,124,125,127,129,132,133,136,138,140,
141,144,145,146,148,150,153,156,159,160,162,165,166,168,
169,170,174,175}

192
{1,3,7,8,9,11,14,16,17,18,22,24,26,28,29,30,34,36,37,39,
43,44,45,47,50,52,53,54,58,60,61,63,67,68,69,71,73,75,79,
80,81,83,86,88,89,90,94,96,98,100,101,102,106,108,109,111,
115,116,117,119,121,123,127,128,129,131,134,136,137,138,
142,144,145,147,151,152,153,155,158,160,161,162,166,168,
170,172,173,174,178,180,181,183,187,188,189,191}

200
{1,4,6,7,8,10,12,16,19,22,24,26,27,28,30,31,32,34,36,38,
40,43,46,48,49,50,52,54,58,60,62,64,66,68,70,72,74,76,78,
80,82,84,85,88,90,91,92,94,98,100,102,104,105,106,108,112,
114,115,118,120,122,124,126,128,130,132,134,136,138,140,
142,144,145,146,148,150,154,156,157,159,160,162,164,166,
168,172,176,178,180,181,183,184,186,187,188,190,192,196,
198,199}

208
{1,2,5,7,11,13,14,17,18,21,22,23,25,28,29,30,32,34,38,40,
42,43,44,46,47,50,55,58,59,61,62,63,65,67,71,73,75,76,77,
80,82,83,84,87,88,91,92,94,98,100,103,104,107,108,110,112,
113,114,116,119,120,123,124,128,130,131,135,137,139,140,
141,143,145,149,152,153,155,156,157,158,160,161,164,168,
170,172,173,174,176,178,182,183,185,189,190,193,194,197,
199,200,201,203,205,206}

216
{1,4,6,8,11,12,13,14,16,18,20,22,26,27,28,32,34,39,40,42,
44,46,47,48,50,52,53,54,56,58,60,62,66,67,68,70,72,74,76,
78,81,82,84,86,88,90,92,96,102,103,104,106,107,108,112,
116,117,118,119,120,122,123,124,126,128,130,132,134,137,
138,140,142,144,146,148,152,153,154,156,158,160,162,166,
168,172,174,176,179,180,181,182,184,186,187,188,192,193,
194,196,198,200,202,207,208,210,212,214,215}

224
{1,3,6,9,10,12,13,15,16,18,21,25,26,28,30,33,34,35,37,38,
40,43,46,49,50,52,53,55,58,60,61,63,64,67,70,73,75,76,78,
79,80,83,85,87,88,92,95,97,98,100,101,103,104,107,110,112,
114,116,117,119,120,123,126,129,131,132,134,135,136,139,
141,143,144,148,151,153,154,156,157,159,160,163,166,168,
169,171,174,177,178,180,181,183,184,186,189,193,194,196,
198,201,202,203,205,206,208,211,214,217,218,220,221,223}

232
{1,4,6,10,11,12,14,15,16,18,20,24,25,27,28,30,34,35,36,
40,42,44,45,46,48,50,52,54,56,62,64,66,69,70,72,74,75,76,
78,80,81,84,86,88,90,91,94,96,99,100,102,104,105,106,108,
110,111,112,114,116,118,120,124,126,130,132,135,136,138,
140,141,144,146,148,150,151,154,156,160,162,165,166,168,
170,172,173,174,175,176,178,180,182,184,186,190,192,194,
195,196,200,201,202,204,207,210,211,212,214,216,220,224,
225,226,228,230,231}

240
{1,6,7,8,9,10,11,12,13,23,26,27,28,29,30,31,32,34,37,38,
42,44,45,46,47,51,53,54,55,60,61,66,67,68,70,74,75,76,77,
79,83,84,87,89,90,91,92,93,94,95,98,108,109,110,111,112,
113,114,115,120,122,123,124,125,134,135,136,137,138,139,
140,141,142,144,145,153,155,156,159,160,161,163,168,169,
170,172,176,177,178,179,182,183,184,185,189,191,192,193,
198,200,201,202,205,206,208,216,217,219,220,221,222,223,
224,225,226,227,236,237,238,239}

248
{1,2,6,8,10,12,14,15,16,20,21,26,28,29,30,32,34,35,36,38,
40,42,44,47,48,50,55,56,57,60,62,64,66,68,70,71,72,76,77,
78,80,84,86,90,91,92,96,98,100,101,102,104,105,106,110,
111,112,114,116,119,120,122,125,126,128,131,132,134,136,
140,141,142,146,150,152,154,155,156,160,161,162,164,166,
167,168,170,174,175,176,180,182,184,186,188,190,191,195,
196,197,198,200,203,204,206,208,210,212,216,218,222,224,
225,226,227,230,231,232,236,238,240,242,244,245,246}

256
{1,4,5,8,10,11,14,15,18,19,22,23,25,28,29,32,34,35,38,39,
41,44,45,48,49,52,53,56,58,59,62,63,66,67,70,71,73,76,77,
80,81,84,85,88,90,91,94,95,97,100,101,104,106,107,110,111,
114,115,118,119,121,124,125,128,130,131,134,135,137,140,
141,144,145,148,149,152,154,155,158,159,161,164,165,168,
170,171,174,175,178,179,182,183,185,188,189,192,193,196,
197,200,202,203,206,207,210,211,214,215,217,220,221,224,
226,227,230,231,233,236,237,240,241,244,245,248,250,251,
254,255}

264
{1,2,4,7,9,12,14,16,19,20,22,24,26,28,30,32,33,36,39,40,
42,44,47,49,50,52,54,56,58,59,60,62,64,65,68,70,72,74,77,
79,80,82,84,86,89,92,94,96,98,100,102,104,105,107,110,112,
114,117,119,120,122,124,126,128,130,132,134,136,138,140,
142,144,147,149,150,152,154,156,157,159,162,164,166,168,
170,172,174,175,177,178,180,182,184,187,189,190,192,194,
196,198,199,202,204,208,210,212,214,217,219,220,222,224,
227,228,230,231,234,236,238,240,242,244,247,248,250,252,
254,255,257,259,260,262}

272
{1,2,8,9,10,11,13,14,17,20,22,23,24,27,30,33,34,35,37,41,
42,44,45,47,48,52,54,55,56,59,62,65,66,67,69,73,74,76,77,
79,80,84,86,87,88,91,94,97,98,99,101,105,106,108,109,111,
112,116,118,119,121,122,125,130,131,132,133,134,139,140,
141,142,143,148,151,152,154,155,157,161,162,164,165,167,
168,172,174,175,176,179,182,185,186,187,189,193,194,196,
197,199,200,204,206,207,208,211,214,217,218,219,221,225,
226,228,229,231,232,236,238,239,240,243,246,249,250,251,
253,256,259,260,262,263,264,265,271,272}

280
{1,3,5,9,11,13,14,16,17,19,23,24,25,26,29,31,33,34,35,39,
41,43,44,45,49,53,54,55,59,61,63,64,65,68,69,71,73,74,79,
80,83,85,86,89,91,93,94,95,99,101,103,104,105,109,110,111,
113,114,115,119,121,123,124,125,129,131,133,134,139,140,
143,144,145,146,149,151,153,154,155,159,161,163,164,165,
169,173,174,175,179,181,183,184,185,189,191,193,194,197,
199,200,203,204,205,206,209,211,214,215,219,221,223,224,
225,229,230,231,233,234,235,239,241,243,244,245,249,251,
253,254,259,260,261,263,266,269,271,273,274,275,277,279}