This is really an addendum to Alex's answer. I wrote a program in SageMath (using GAP) that computes these numbers, so I was able to expand Alex's lists considerably. Each of these lists should be complete in the sense that there are no missing numbers less than the last number on the list.
$\aleph_2$: 1, 3, 5, 6, 7, 10, 12, 15, 18, 19, 20, 21, 22, 25, 27, 28, 31, 35, 36, 42, 43, 45, 46, 47, 48, 50, 51, 54, 55, 56, 60, 65, 66, 67, 68, 70, 73, 74, 75, 78, 79, 83, 84, 85, 90, 91,...
$\aleph_3$: 1, 4, 7, 10, 13, 20, 21, 22, 34, 35, 37, 46, 49, 50, 52, 56, 63, 64, 65, 84, 91, 95, 97, 99, 105, 106, 120, 140, 155, 157, 161, 164, 165, 174, 175, 183, 204, 208, 210, 211, 220, 223, 230, 232, 234, 252, 260, 264, 266, 270, 285, 286,...
$\aleph_4$: 1, 5, 9, 15, 21, 35, 39, 61, 65, 70, 71, 95, 99, 125, 126, 129, 166, 189, 210, 215, 231, 241, 291, 325, 330, 369, 380, 401, 406, 411, 420, 421, 435, 459, 460, 465, 495, 545, 555, 560, 571, 624, 666, 700, 701, 714, 715, 725, 729, 735, 741, 744, 765, 795, 825, 909, 924, 1001,...
$\aleph_5$: 1, 6, 11, 21, 31, 51, 56, 66, 91, 111, 126, 131, 161, 171, 176, 216, 231, 252, 253, 261, 341, 377, 381, 406, 462, 471, 511, 651, 671, 693, 786, 792, 851, 882, 913, 921, 931, 966, 1091, 1111, 1121, 1141, 1191, 1231, 1287, 1426, 1521, 1551, 1596, 1611, 1722, 1728, 1767, 1771, 1836, 1891, 1911, 1966, 2002, 2156, 2211, 2371, 2522, 2661, 2681, 2761, 2786, 3003,...
$\aleph_6$: 1, 7, 13, 28, 43, 70, 84, 104, 127, 175, 210, 225, 252, 280, 286, 343, 377, 462, 468, 490, 559, 616, 714, 777, 869, 924, 925, 967, 1105, 1176, 1456, 1469, 1611, 1716, 1722, 1736, 1807, 1833, 1974, 2100, 2289, 2296, 2353, 2379, 2394, 2842, 3003, 3095, 3164, 3234, 3380, 3696, 3739, 3744, 3892, 3899, 3920, 4214, 4235, 4439, 4579, 4745, 5005, 5068, 5629, 5740, 5754, 6076, 6201, 6237, 6517, 6929, 7260, 7267, 7644, 7657, 7995, 8008,...
The $\aleph_2$ list is actually considerably harder to compute than the others, which are all roughly comparable, so there's no particular reason I stopped at $\aleph_6$.