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Corrected and extended list of examples.
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Stefan Kohl ♦
Stefan Kohl ♦
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the average value of $P(n)/n$ is asymptotically $\zeta(2)/\log n$, which tends to $0$ when $n$ tends to $\infty$. Therefore, regardless of whether you require $q^2$, $q^4$, $q^{10}$ or some higher power of $q$ to divide $n!$, the natural density of the set of the $n$ which fulfil your condition equals $1$. As already said in the comments, some examples are $n = 538$ ($q^2 | n!$), $n = 1342$ ($q^6|n!$) and $n = 9712$ ($q^8|n!$).

.

A list of examples $(n,q,s)$ where $n \leq 10000$ is as follows:

[ [ 539, 269, 2 ], [ 540, 269, 2 ], [ 903, 449, 2 ], [ 904, 449, 2 ],
  [ 905, 449, 2 ], [ 906, 449, 2 ], [ 1085, 541, 2 ], [ 1086, 541, 2 ],
  [ 1145, 571, 2 ], [ 1146, 571, 2 ], [ 1147, 571, 2 ], [ 1148, 571, 2 ],
  [ 1149, 571, 2 ], [ 1150, 571, 2 ], [ 1275, 631, 2 ], [ 1276, 631, 2 ],
  [ 1343, 443, 3 ], [ 1344, 443, 3 ], [ 1345, 269, 5 ], [ 1346, 673, 2 ],
  [ 1347, 673, 2 ], [ 1348, 673, 2 ], [ 1349, 673, 2 ], [ 1350, 673, 2 ],
  [ 1351, 673, 2 ], [ 1352, 673, 2 ], [ 1353, 673, 2 ], [ 1354, 677, 2 ],
  [ 1355, 677, 2 ], [ 1356, 677, 2 ], [ 1357, 677, 2 ], [ 1358, 677, 2 ],
  [ 1359, 677, 2 ], [ 1360, 677, 2 ], [ 1397, 691, 2 ], [ 1398, 463, 3 ],
  [ 1653, 823, 2 ], [ 1654, 827, 2 ], [ 1655, 827, 2 ], [ 1656, 827, 2 ],
  [ 1685, 839, 2 ], [ 1686, 839, 2 ], [ 1687, 839, 2 ], [ 1688, 839, 2 ],
  [ 1689, 839, 2 ], [ 1690, 839, 2 ], [ 1691, 839, 2 ], [ 1692, 839, 2 ],
  [ 1775, 887, 2 ], [ 1776, 887, 2 ], [ 1929, 643, 3 ], [ 1930, 643, 3 ],
  [ 1967, 983, 2 ], [ 1968, 983, 2 ], [ 1969, 983, 2 ], [ 1970, 983, 2 ],
  [ 1971, 983, 2 ], [ 1972, 983, 2 ], [ 2177, 1087, 2 ], [ 2178, 1087, 2 ],
  [ 2195, 1097, 2 ], [ 2196, 1097, 2 ], [ 2197, 1097, 2 ], [ 2198, 1097, 2 ],
  [ 2199, 1097, 2 ], [ 2200, 1097, 2 ], [ 2201, 1097, 2 ], [ 2202, 1097, 2 ],
  [ 2327, 1163, 2 ], [ 2328, 1163, 2 ], [ 2329, 1163, 2 ], [ 2330, 1163, 2 ],
  [ 2331, 1163, 2 ], [ 2332, 1163, 2 ], [ 2493, 829, 3 ], [ 2494, 829, 3 ],
  [ 2495, 829, 3 ], [ 2496, 829, 3 ], [ 2497, 829, 3 ], [ 2498, 1249, 2 ],
  [ 2499, 1249, 2 ], [ 2500, 1249, 2 ], [ 2501, 1249, 2 ], [ 2502, 1249, 2 ],
  [ 2519, 1259, 2 ], [ 2520, 1259, 2 ], [ 2573, 1283, 2 ], [ 2574, 1283, 2 ],
  [ 2575, 1283, 2 ], [ 2576, 1283, 2 ], [ 2577, 1283, 2 ], [ 2578, 1289, 2 ],
  [ 2877, 1433, 2 ], [ 2878, 1439, 2 ], [ 2987, 1493, 2 ], [ 2988, 1493, 2 ],
  [ 2989, 1493, 2 ], [ 2990, 1493, 2 ], [ 2991, 1493, 2 ], [ 2992, 1493, 2 ],
  [ 2993, 1493, 2 ], [ 2994, 1493, 2 ], [ 2995, 1493, 2 ], [ 2996, 1493, 2 ],
  [ 2997, 1493, 2 ], [ 2998, 1499, 2 ], [ 3105, 1549, 2 ], [ 3106, 1553, 2 ],
  [ 3107, 1553, 2 ], [ 3108, 1553, 2 ], [ 3153, 1571, 2 ], [ 3154, 1571, 2 ],
  [ 3155, 1571, 2 ], [ 3156, 1571, 2 ], [ 3157, 1571, 2 ], [ 3158, 1579, 2 ],
  [ 3159, 1579, 2 ], [ 3160, 1579, 2 ], [ 3161, 1579, 2 ], [ 3162, 1579, 2 ],
  [ 3245, 1621, 2 ], [ 3246, 1621, 2 ], [ 3247, 1621, 2 ], [ 3248, 1621, 2 ],
  [ 3249, 1621, 2 ], [ 3250, 1621, 2 ], [ 3287, 1637, 2 ], [ 3288, 1637, 2 ],
  [ 3289, 1637, 2 ], [ 3290, 1093, 3 ], [ 3291, 1097, 3 ], [ 3292, 1097, 3 ],
  [ 3293, 1097, 3 ], [ 3294, 1097, 3 ], [ 3295, 1097, 3 ], [ 3296, 1097, 3 ],
  [ 3297, 1097, 3 ], [ 3298, 1097, 3 ], [ 3429, 1709, 2 ], [ 3430, 1709, 2 ],
  [ 3431, 1709, 2 ], [ 3432, 1709, 2 ], [ 3485, 1741, 2 ], [ 3486, 1741, 2 ],
  [ 3487, 1741, 2 ], [ 3488, 1741, 2 ], [ 3489, 1741, 2 ], [ 3490, 1741, 2 ],
  [ 3755, 1877, 2 ], [ 3756, 1877, 2 ], [ 3757, 1877, 2 ], [ 3758, 1879, 2 ],
  [ 3759, 1879, 2 ], [ 3760, 1879, 2 ], [ 3819, 1907, 2 ], [ 3820, 1907, 2 ],
  [ 3905, 1951, 2 ], [ 3906, 1951, 2 ], [ 3963, 1979, 2 ], [ 3964, 1979, 2 ],
  [ 3965, 1979, 2 ], [ 3966, 1979, 2 ], [ 3983, 1987, 2 ], [ 3984, 1987, 2 ],
  [ 3985, 1987, 2 ], [ 3986, 1993, 2 ], [ 3987, 1993, 2 ], [ 3988, 1993, 2 ],
  [ 4043, 2017, 2 ], [ 4044, 2017, 2 ], [ 4045, 2017, 2 ], [ 4046, 2017, 2 ],
  [ 4047, 2017, 2 ], [ 4048, 2017, 2 ], [ 4175, 2087, 2 ], [ 4176, 2087, 2 ],
  [ 4193, 2089, 2 ], [ 4194, 599, 7 ], [ 4195, 839, 5 ], [ 4196, 1049, 4 ],
  [ 4197, 1399, 3 ], [ 4198, 2099, 2 ], [ 4199, 2099, 2 ], [ 4200, 2099, 2 ],
  [ 4313, 2153, 2 ], [ 4314, 2153, 2 ], [ 4315, 2153, 2 ], [ 4316, 2153, 2 ],
  [ 4317, 2153, 2 ], [ 4318, 2153, 2 ], [ 4319, 2153, 2 ], [ 4320, 2153, 2 ],
  [ 4321, 2153, 2 ], [ 4322, 2161, 2 ], [ 4323, 2161, 2 ], [ 4324, 2161, 2 ],
  [ 4325, 2161, 2 ], [ 4326, 2161, 2 ], [ 4389, 1459, 3 ], [ 4390, 1459, 3 ],
  [ 4439, 2213, 2 ], [ 4440, 2213, 2 ], [ 4479, 2239, 2 ], [ 4480, 2239, 2 ],
  [ 4539, 2269, 2 ], [ 4540, 2269, 2 ], [ 4541, 2269, 2 ], [ 4542, 2269, 2 ],
  [ 4543, 2269, 2 ], [ 4544, 2269, 2 ], [ 4545, 2269, 2 ], [ 4546, 2273, 2 ],
  [ 4619, 2309, 2 ], [ 4620, 2309, 2 ], [ 4719, 2357, 2 ], [ 4720, 2357, 2 ],
  [ 4749, 2371, 2 ], [ 4750, 2371, 2 ], [ 4775, 2383, 2 ], [ 4776, 2383, 2 ],
  [ 4777, 2383, 2 ], [ 4778, 2389, 2 ], [ 4779, 2389, 2 ], [ 4780, 2389, 2 ],
  [ 4781, 2389, 2 ], [ 4782, 2389, 2 ], [ 4847, 2423, 2 ], [ 4848, 2423, 2 ],
  [ 4849, 2423, 2 ], [ 4850, 2423, 2 ], [ 4851, 2423, 2 ], [ 4852, 2423, 2 ],
  [ 4853, 2423, 2 ], [ 4854, 2423, 2 ], [ 4855, 2423, 2 ], [ 4856, 2423, 2 ],
  [ 4857, 2423, 2 ], [ 4858, 2423, 2 ], [ 4859, 2423, 2 ], [ 4860, 2423, 2 ],
  [ 5075, 2531, 2 ], [ 5076, 2531, 2 ], [ 5135, 1709, 3 ], [ 5136, 1709, 3 ],
  [ 5137, 1709, 3 ], [ 5138, 1709, 3 ], [ 5139, 1709, 3 ], [ 5140, 1709, 3 ],
  [ 5141, 1709, 3 ], [ 5142, 1709, 3 ], [ 5143, 1283, 4 ], [ 5144, 1283, 4 ],
  [ 5145, 1283, 4 ], [ 5146, 1283, 4 ], [ 5225, 2609, 2 ], [ 5226, 2609, 2 ],
  [ 5253, 2621, 2 ], [ 5254, 2621, 2 ], [ 5255, 2621, 2 ], [ 5256, 2621, 2 ],
  [ 5257, 2621, 2 ], [ 5258, 1051, 5 ], [ 5259, 1753, 3 ], [ 5260, 1753, 3 ],
  [ 5367, 2683, 2 ], [ 5368, 2683, 2 ], [ 5369, 2683, 2 ], [ 5370, 2683, 2 ],
  [ 5371, 2683, 2 ], [ 5372, 2683, 2 ], [ 5373, 2683, 2 ], [ 5374, 2687, 2 ],
  [ 5375, 2687, 2 ], [ 5376, 2687, 2 ], [ 5377, 2687, 2 ], [ 5378, 2689, 2 ],
  [ 5379, 2689, 2 ], [ 5380, 2689, 2 ], [ 5465, 2731, 2 ], [ 5466, 2731, 2 ],
  [ 5467, 2731, 2 ], [ 5468, 2731, 2 ], [ 5469, 2731, 2 ], [ 5470, 2731, 2 ],
  [ 5499, 2749, 2 ], [ 5500, 2749, 2 ], [ 5547, 2767, 2 ], [ 5548, 2767, 2 ],
  [ 5549, 2767, 2 ], [ 5550, 1847, 3 ], [ 5551, 1847, 3 ], [ 5552, 1847, 3 ],
  [ 5553, 1847, 3 ], [ 5554, 2777, 2 ], [ 5555, 2777, 2 ], [ 5556, 2777, 2 ],
  [ 5607, 2803, 2 ], [ 5608, 2803, 2 ], [ 5609, 2803, 2 ], [ 5610, 2803, 2 ],
  [ 5611, 2803, 2 ], [ 5612, 2803, 2 ], [ 5613, 2803, 2 ], [ 5614, 2803, 2 ],
  [ 5615, 2803, 2 ], [ 5616, 2803, 2 ], [ 5617, 2803, 2 ], [ 5618, 2803, 2 ],
  [ 5619, 2803, 2 ], [ 5620, 2803, 2 ], [ 5621, 2803, 2 ], [ 5622, 1873, 3 ],
  [ 5733, 2861, 2 ], [ 5734, 2861, 2 ], [ 5735, 2861, 2 ], [ 5736, 2861, 2 ],
  [ 5765, 2879, 2 ], [ 5766, 2879, 2 ], [ 5767, 2879, 2 ], [ 5768, 2879, 2 ],
  [ 5769, 2879, 2 ], [ 5770, 2879, 2 ], [ 5771, 2879, 2 ], [ 5772, 2879, 2 ],
  [ 5773, 2879, 2 ], [ 5774, 2887, 2 ], [ 5775, 2887, 2 ], [ 5776, 2887, 2 ],
  [ 5777, 2887, 2 ], [ 5778, 2887, 2 ], [ 5919, 2957, 2 ], [ 5920, 2957, 2 ],
  [ 5921, 2957, 2 ], [ 5922, 2957, 2 ], [ 5969, 1987, 3 ], [ 5970, 1987, 3 ],
  [ 5971, 1987, 3 ], [ 5972, 1987, 3 ], [ 5973, 1987, 3 ], [ 5974, 1987, 3 ],
  [ 5975, 1987, 3 ], [ 5976, 1987, 3 ], [ 5977, 1493, 4 ], [ 5978, 1493, 4 ],
  [ 5979, 1993, 3 ], [ 5980, 1993, 3 ], [ 6003, 3001, 2 ], [ 6004, 3001, 2 ],
  [ 6005, 3001, 2 ], [ 6006, 3001, 2 ], [ 6027, 3011, 2 ], [ 6028, 3011, 2 ],
  [ 6189, 3089, 2 ], [ 6190, 3089, 2 ], [ 6191, 3089, 2 ], [ 6192, 3089, 2 ],
  [ 6193, 3089, 2 ], [ 6194, 2063, 3 ], [ 6195, 2063, 3 ], [ 6196, 2063, 3 ],
  [ 6245, 3121, 2 ], [ 6246, 3121, 2 ], [ 6413, 3203, 2 ], [ 6414, 3203, 2 ],
  [ 6415, 3203, 2 ], [ 6416, 3203, 2 ], [ 6417, 3203, 2 ], [ 6418, 3209, 2 ],
  [ 6419, 3209, 2 ], [ 6420, 3209, 2 ], [ 6443, 3221, 2 ], [ 6444, 3221, 2 ],
  [ 6445, 3221, 2 ], [ 6446, 3221, 2 ], [ 6447, 3221, 2 ], [ 6448, 3221, 2 ],
  [ 6467, 3229, 2 ], [ 6468, 3229, 2 ], [ 6507, 3253, 2 ], [ 6508, 3253, 2 ],
  [ 6509, 3253, 2 ], [ 6510, 3253, 2 ], [ 6511, 3253, 2 ], [ 6512, 3253, 2 ],
  [ 6513, 3253, 2 ], [ 6514, 3257, 2 ], [ 6515, 3257, 2 ], [ 6516, 3257, 2 ],
  [ 6517, 3257, 2 ], [ 6518, 3259, 2 ], [ 6519, 3259, 2 ], [ 6520, 3259, 2 ],
  [ 6545, 3271, 2 ], [ 6546, 3271, 2 ], [ 6597, 1319, 5 ], [ 6598, 3299, 2 ],
  [ 6635, 3313, 2 ], [ 6636, 3313, 2 ], [ 6753, 3373, 2 ], [ 6754, 3373, 2 ],
  [ 6755, 3373, 2 ], [ 6756, 3373, 2 ], [ 6757, 3373, 2 ], [ 6758, 3373, 2 ],
  [ 6759, 3373, 2 ], [ 6760, 3373, 2 ], [ 6819, 3407, 2 ], [ 6820, 3407, 2 ],
  [ 6821, 3407, 2 ], [ 6822, 3407, 2 ], [ 6933, 3463, 2 ], [ 6934, 3467, 2 ],
  [ 6935, 3467, 2 ], [ 6936, 3467, 2 ], [ 6937, 3467, 2 ], [ 6938, 3469, 2 ],
  [ 6939, 3469, 2 ], [ 6940, 3469, 2 ], [ 6941, 3469, 2 ], [ 6942, 3469, 2 ],
  [ 6943, 3469, 2 ], [ 6944, 3469, 2 ], [ 6945, 3469, 2 ], [ 6946, 3469, 2 ],
  [ 7095, 3547, 2 ], [ 7096, 3547, 2 ], [ 7097, 3547, 2 ], [ 7098, 3547, 2 ],
  [ 7099, 3547, 2 ], [ 7100, 3547, 2 ], [ 7101, 3547, 2 ], [ 7102, 3547, 2 ],
  [ 7145, 3571, 2 ], [ 7146, 3571, 2 ], [ 7147, 3571, 2 ], [ 7148, 3571, 2 ],
  [ 7149, 3571, 2 ], [ 7150, 3571, 2 ], [ 7175, 3583, 2 ], [ 7176, 3583, 2 ],
  [ 7269, 3631, 2 ], [ 7270, 3631, 2 ], [ 7271, 3631, 2 ], [ 7272, 3631, 2 ],
  [ 7273, 3631, 2 ], [ 7274, 3637, 2 ], [ 7275, 3637, 2 ], [ 7276, 3637, 2 ],
  [ 7277, 3637, 2 ], [ 7278, 3637, 2 ], [ 7279, 3637, 2 ], [ 7280, 3637, 2 ],
  [ 7281, 3637, 2 ], [ 7282, 3637, 2 ], [ 7367, 3677, 2 ], [ 7368, 3677, 2 ],
  [ 7385, 3691, 2 ], [ 7386, 3691, 2 ], [ 7387, 3691, 2 ], [ 7388, 3691, 2 ],
  [ 7389, 3691, 2 ], [ 7390, 3691, 2 ], [ 7391, 3691, 2 ], [ 7392, 3691, 2 ],
  [ 7409, 3701, 2 ], [ 7410, 3701, 2 ], [ 7449, 3719, 2 ], [ 7450, 3719, 2 ],
  [ 7475, 3733, 2 ], [ 7476, 3733, 2 ], [ 7637, 2543, 3 ], [ 7638, 2543, 3 ],
  [ 7665, 2551, 3 ], [ 7666, 3833, 2 ], [ 7667, 3833, 2 ], [ 7668, 3833, 2 ],
  [ 7775, 3881, 2 ], [ 7776, 3881, 2 ], [ 7777, 3881, 2 ], [ 7778, 3889, 2 ],
  [ 7779, 3889, 2 ], [ 7780, 3889, 2 ], [ 7781, 3889, 2 ], [ 7782, 3889, 2 ],
  [ 7783, 3889, 2 ], [ 7784, 3889, 2 ], [ 7785, 3889, 2 ], [ 7786, 3889, 2 ],
  [ 7787, 3889, 2 ], [ 7788, 3889, 2 ], [ 7809, 1951, 4 ], [ 7810, 1951, 4 ],
  [ 7811, 1951, 4 ], [ 7812, 1951, 4 ], [ 7813, 1951, 4 ], [ 7814, 3907, 2 ],
  [ 7815, 3907, 2 ], [ 7816, 3907, 2 ], [ 7899, 3947, 2 ], [ 7900, 3947, 2 ],
  [ 7979, 3989, 2 ], [ 7980, 3989, 2 ], [ 7981, 3989, 2 ], [ 7982, 3989, 2 ],
  [ 7983, 3989, 2 ], [ 7984, 3989, 2 ], [ 7985, 3989, 2 ], [ 7986, 3989, 2 ],
  [ 7987, 3989, 2 ], [ 7988, 3989, 2 ], [ 7989, 3989, 2 ], [ 7990, 3989, 2 ],
  [ 7991, 3989, 2 ], [ 7992, 3989, 2 ], [ 8033, 4013, 2 ], [ 8034, 4013, 2 ],
  [ 8035, 4013, 2 ], [ 8036, 4013, 2 ], [ 8037, 4013, 2 ], [ 8038, 4019, 2 ],
  [ 8139, 2713, 3 ], [ 8140, 2713, 3 ], [ 8141, 2713, 3 ], [ 8142, 2713, 3 ],
  [ 8143, 2713, 3 ], [ 8144, 2713, 3 ], [ 8145, 2713, 3 ], [ 8146, 4073, 2 ],
  [ 8207, 4099, 2 ], [ 8208, 4099, 2 ], [ 8259, 4129, 2 ], [ 8260, 4129, 2 ],
  [ 8261, 4129, 2 ], [ 8262, 4129, 2 ], [ 8345, 2777, 3 ], [ 8346, 2777, 3 ],
  [ 8347, 2083, 4 ], [ 8348, 2087, 4 ], [ 8349, 2087, 4 ], [ 8350, 2087, 4 ],
  [ 8351, 2087, 4 ], [ 8352, 2087, 4 ], [ 8405, 4201, 2 ], [ 8406, 4201, 2 ],
  [ 8407, 4201, 2 ], [ 8408, 4201, 2 ], [ 8409, 4201, 2 ], [ 8410, 4201, 2 ],
  [ 8411, 4201, 2 ], [ 8412, 4201, 2 ], [ 8413, 4201, 2 ], [ 8414, 4201, 2 ],
  [ 8415, 4201, 2 ], [ 8416, 4201, 2 ], [ 8417, 4201, 2 ], [ 8418, 2803, 3 ],
  [ 8483, 4241, 2 ], [ 8484, 4241, 2 ], [ 8485, 4241, 2 ], [ 8486, 4243, 2 ],
  [ 8487, 4243, 2 ], [ 8488, 4243, 2 ], [ 8489, 4243, 2 ], [ 8490, 4243, 2 ],
  [ 8491, 4243, 2 ], [ 8492, 4243, 2 ], [ 8493, 4243, 2 ], [ 8494, 4243, 2 ],
  [ 8495, 4243, 2 ], [ 8496, 4243, 2 ], [ 8497, 4243, 2 ], [ 8498, 4243, 2 ],
  [ 8499, 4243, 2 ], [ 8500, 4243, 2 ], [ 8559, 4273, 2 ], [ 8560, 4273, 2 ],
  [ 8561, 4273, 2 ], [ 8562, 2851, 3 ], [ 8777, 1753, 5 ], [ 8778, 1753, 5 ],
  [ 8799, 4397, 2 ], [ 8800, 4397, 2 ], [ 8801, 4397, 2 ], [ 8802, 4397, 2 ],
  [ 8883, 4441, 2 ], [ 8884, 4441, 2 ], [ 8885, 4441, 2 ], [ 8886, 4441, 2 ],
  [ 8909, 4451, 2 ], [ 8910, 4451, 2 ], [ 8911, 4451, 2 ], [ 8912, 4451, 2 ],
  [ 8913, 4451, 2 ], [ 8914, 4457, 2 ], [ 8915, 4457, 2 ], [ 8916, 4457, 2 ],
  [ 8917, 4457, 2 ], [ 8918, 4457, 2 ], [ 8919, 4457, 2 ], [ 8920, 4457, 2 ],
  [ 8921, 4457, 2 ], [ 8922, 4457, 2 ], [ 8987, 4493, 2 ], [ 8988, 4493, 2 ],
  [ 8989, 4493, 2 ], [ 8990, 4493, 2 ], [ 8991, 4493, 2 ], [ 8992, 4493, 2 ],
  [ 8993, 4493, 2 ], [ 8994, 4493, 2 ], [ 8995, 4493, 2 ], [ 8996, 4493, 2 ],
  [ 8997, 4493, 2 ], [ 8998, 4493, 2 ], [ 9083, 3023, 3 ], [ 9084, 3023, 3 ],
  [ 9085, 2269, 4 ], [ 9086, 2269, 4 ], [ 9087, 2269, 4 ], [ 9088, 2269, 4 ],
  [ 9089, 2269, 4 ], [ 9090, 2269, 4 ], [ 9125, 4561, 2 ], [ 9126, 4561, 2 ],
  [ 9273, 3089, 3 ], [ 9274, 4637, 2 ], [ 9275, 4637, 2 ], [ 9276, 4637, 2 ],
  [ 9309, 4651, 2 ], [ 9310, 4651, 2 ], [ 9365, 4679, 2 ], [ 9366, 4679, 2 ],
  [ 9367, 4679, 2 ], [ 9368, 4679, 2 ], [ 9369, 4679, 2 ], [ 9370, 4679, 2 ],
  [ 9455, 4723, 2 ], [ 9456, 4723, 2 ], [ 9457, 4723, 2 ], [ 9458, 4729, 2 ],
  [ 9459, 4729, 2 ], [ 9460, 4729, 2 ], [ 9567, 4783, 2 ], [ 9568, 4783, 2 ],
  [ 9569, 4783, 2 ], [ 9570, 4783, 2 ], [ 9571, 4783, 2 ], [ 9572, 4783, 2 ],
  [ 9573, 4783, 2 ], [ 9574, 4787, 2 ], [ 9575, 4787, 2 ], [ 9576, 4787, 2 ],
  [ 9577, 4787, 2 ], [ 9578, 4789, 2 ], [ 9579, 4789, 2 ], [ 9580, 4789, 2 ],
  [ 9581, 4789, 2 ], [ 9582, 4789, 2 ], [ 9583, 4789, 2 ], [ 9584, 4789, 2 ],
  [ 9585, 4789, 2 ], [ 9586, 4793, 2 ], [ 9713, 1213, 8 ], [ 9714, 1619, 6 ],
  [ 9715, 1619, 6 ], [ 9716, 1619, 6 ], [ 9717, 1619, 6 ], [ 9718, 1619, 6 ],
  [ 9765, 4877, 2 ], [ 9766, 4877, 2 ], [ 9965, 3319, 3 ], [ 9966, 3319, 3 ],
  [ 9989, 4993, 2 ], [ 9990, 4993, 2 ], [ 9991, 4993, 2 ], [ 9992, 4993, 2 ],
  [ 9993, 4993, 2 ], [ 9994, 4993, 2 ], [ 9995, 4993, 2 ], [ 9996, 4993, 2 ],
  [ 9997, 4993, 2 ], [ 9998, 4999, 2 ], [ 9999, 4999, 2 ], [ 10000, 4999, 2 ]

the average value of $P(n)/n$ is asymptotically $\zeta(2)/\log n$, which tends to $0$ when $n$ tends to $\infty$. Therefore, regardless of whether you require $q^2$, $q^4$, $q^{10}$ or some higher power of $q$ to divide $n!$, the natural density of the set of the $n$ which fulfil your condition equals $1$. As already said in the comments, some examples are $n = 538$ ($q^2 | n!$), $n = 1342$ ($q^6|n!$) and $n = 9712$ ($q^8|n!$).

the average value of $P(n)/n$ is asymptotically $\zeta(2)/\log n$, which tends to $0$ when $n$ tends to $\infty$. Therefore, regardless of whether you require $q^2$, $q^4$, $q^{10}$ or some higher power of $q$ to divide $n!$, the natural density of the set of the $n$ which fulfil your condition equals $1$.

A list of examples $(n,q,s)$ where $n \leq 10000$ is as follows:

[ [ 539, 269, 2 ], [ 540, 269, 2 ], [ 903, 449, 2 ], [ 904, 449, 2 ],
  [ 905, 449, 2 ], [ 906, 449, 2 ], [ 1085, 541, 2 ], [ 1086, 541, 2 ],
  [ 1145, 571, 2 ], [ 1146, 571, 2 ], [ 1147, 571, 2 ], [ 1148, 571, 2 ],
  [ 1149, 571, 2 ], [ 1150, 571, 2 ], [ 1275, 631, 2 ], [ 1276, 631, 2 ],
  [ 1343, 443, 3 ], [ 1344, 443, 3 ], [ 1345, 269, 5 ], [ 1346, 673, 2 ],
  [ 1347, 673, 2 ], [ 1348, 673, 2 ], [ 1349, 673, 2 ], [ 1350, 673, 2 ],
  [ 1351, 673, 2 ], [ 1352, 673, 2 ], [ 1353, 673, 2 ], [ 1354, 677, 2 ],
  [ 1355, 677, 2 ], [ 1356, 677, 2 ], [ 1357, 677, 2 ], [ 1358, 677, 2 ],
  [ 1359, 677, 2 ], [ 1360, 677, 2 ], [ 1397, 691, 2 ], [ 1398, 463, 3 ],
  [ 1653, 823, 2 ], [ 1654, 827, 2 ], [ 1655, 827, 2 ], [ 1656, 827, 2 ],
  [ 1685, 839, 2 ], [ 1686, 839, 2 ], [ 1687, 839, 2 ], [ 1688, 839, 2 ],
  [ 1689, 839, 2 ], [ 1690, 839, 2 ], [ 1691, 839, 2 ], [ 1692, 839, 2 ],
  [ 1775, 887, 2 ], [ 1776, 887, 2 ], [ 1929, 643, 3 ], [ 1930, 643, 3 ],
  [ 1967, 983, 2 ], [ 1968, 983, 2 ], [ 1969, 983, 2 ], [ 1970, 983, 2 ],
  [ 1971, 983, 2 ], [ 1972, 983, 2 ], [ 2177, 1087, 2 ], [ 2178, 1087, 2 ],
  [ 2195, 1097, 2 ], [ 2196, 1097, 2 ], [ 2197, 1097, 2 ], [ 2198, 1097, 2 ],
  [ 2199, 1097, 2 ], [ 2200, 1097, 2 ], [ 2201, 1097, 2 ], [ 2202, 1097, 2 ],
  [ 2327, 1163, 2 ], [ 2328, 1163, 2 ], [ 2329, 1163, 2 ], [ 2330, 1163, 2 ],
  [ 2331, 1163, 2 ], [ 2332, 1163, 2 ], [ 2493, 829, 3 ], [ 2494, 829, 3 ],
  [ 2495, 829, 3 ], [ 2496, 829, 3 ], [ 2497, 829, 3 ], [ 2498, 1249, 2 ],
  [ 2499, 1249, 2 ], [ 2500, 1249, 2 ], [ 2501, 1249, 2 ], [ 2502, 1249, 2 ],
  [ 2519, 1259, 2 ], [ 2520, 1259, 2 ], [ 2573, 1283, 2 ], [ 2574, 1283, 2 ],
  [ 2575, 1283, 2 ], [ 2576, 1283, 2 ], [ 2577, 1283, 2 ], [ 2578, 1289, 2 ],
  [ 2877, 1433, 2 ], [ 2878, 1439, 2 ], [ 2987, 1493, 2 ], [ 2988, 1493, 2 ],
  [ 2989, 1493, 2 ], [ 2990, 1493, 2 ], [ 2991, 1493, 2 ], [ 2992, 1493, 2 ],
  [ 2993, 1493, 2 ], [ 2994, 1493, 2 ], [ 2995, 1493, 2 ], [ 2996, 1493, 2 ],
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Stefan Kohl ♦
Stefan Kohl ♦
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Given a positive integer $n$, let $P(n)$ denote the largest prime factor of $n$. What you are looking for are integers $n$ such that $q := \max\{P(n-15),P(n-14),\dots,P(n)\} < Cn$ for some constant $C \in ]0,1[$ depending on whether you require $q^2$ to divide $n!$ (as in the first version of your question) or $q^4$ like now or some higher power. However by

John G. Kemeny: Largest prime factor, Journal of Pure and Applied Algebra 89(1993) Issues 1-2, 181-186

the average value of $P(n)/n$ is asymptotically $\zeta(2)/\log n$, which tends to $0$ when $n$ tends to $\infty$. Therefore, regardless of whether you require $q^2$, $q^4$, $q^{10}$ or some higher power of $q$ to divide $n!$, the natural density of the set of the $n$ which fulfil your condition equals $1$. As already said in the comments, some examples are $n = 538$ ($q^2 | n!$), $n = 1342$ ($q^6|n!$) and $n = 9712$ ($q^8|n!$).