A BE-number such that for all of its representations the permutations bases-exponents are cyclic is called Irreducible.

Irreducibles (up to 10⁶):   A387302

4, 17, 27, 32, 57, 89, 100, 105, 145, 166, 177, 254, 256, 276, 289, 320, 348, 368, 377, 480, 548, 568, 593, 673, 730, 739, 773, 777, 845, 865, 892, 922, 932, 945, 1036, 1065, 1124, 1138, 1164, 1174, 1230, 1236, 1250, 1338, 1355, 1376, 1410, 1433, 1486, 1509, 1533, 1601, 1649, 1652, 1692, 1898, 1930, 2006, 2033, 2049, 2089, 2121, 2125, 2169, 2198, 2244, 2292, 2316, 2371, 2393, 2409, 2512, 2516, 2518, 2530, 2545, 2602, 2789, 2809, 2861, 2908, 2957, 3004, 3082, 3125, 3154, 3174, 3282, 3358, 3388, 3401, 3476, 3477, 3545, 3578, 3647, 3660, 3687, 3842, 4065, 4144, 4148, 4221, 4240, 4260, 4263, 4315, 4337, 4368, 4370, 4393, 4425, 4443, 4509, 4596, 4612, 4652, 4684, 4684, 4745, 4761, 4789, 4884, 4978, 4981, 5008, 5098, 5180, 5181, 5372, 5392, 5401, 5420, 5454, 5471, 5510, 5615, 5645, 5737, 5768, 5833, 5845, 5917, 5921, 6092, 6237, 6241, 6266, 6405, 6453, 6564, 6589, 6628, 6633, 6652, 6661, 6705, 6708, 6782, 6850, 6878, 6905, 6954, 7024, 7073, 7089, 7096, 7109, 7146, 7205, 7346, 7405, 7592, 7661, 7737, 7790, 7865, 7865, 7866, 7902, 7974, 7983, 8049, 8060, 8062, 8122, 8231, 8361, 8379, 8481, 8484, 8538, 8538, 8610, 8630, 8649, 8673, 8728, 8877, 8898, 8900, 8932, 8987, 9001, 9016, 9134, 9154, 9194, 9218, 9298, 9373, 9488, 9617, 9771, 9914, 9961, 10004, 10037, 10201, 10303, 10387, 10398, 10428, 10458, 10486, 10554, 10657, 10681, 10721, 10744, 10765, 10824, 10830, 10900, 10953, 10953, 11040, 11097, 11123, 11144, 11154, 11170, 11205, 11273, 11356, 11394, 11461, 11519, 11594, 11653, 11714, 11806, 11853, 11909, 11913, 11914, 11986, 11986, 12022, 12073, 12078, 12099, 12177, 12182, 12430, 12449, 12449, 12497, 12525, 12569, 12625, 12697, 12746, 12938, 12946, 12947, 13219, 13324, 13463, 13466, 13621, 13730, 13762, 14002, 14052, 14249, 14354, 14409, 14426, 14535, 14544, 14590, 14602, 14797, 14847, 14946, 14969, 14970, 14993, 15085, 15154, 15156, 15260, 15345, 15357, 15389, 15604, 15667, 15693, 15882, 15912, 15970, 15984, 16026, 16030, 16084, 16249, 16358, 16375, 16449, 16493, 16497, 16522, 16580, 16585, 16671, 16672, 16701, 16705, 16752, 16762, 16774, 16794, 16808, 16813, 16890, 16946, 16960, 16969, 17011, 17014, 17021, 17026, 17069, 17090, 17145, 17157, 17188, 17228, 17237, 17253, 17282, 17309, 17393, 17508, 17518, 17530, 17534, 17572, 17576, 17581, 17595, 17633, 17651, 17713, 17726, 17738, 17755, 17793, 17802, 17825, 17847, 17928, 17945, 17948, 18007, 18010, 18069, 18096, 18109, 18149, 18168, 18355, 18389, 18421, 18466, 18497, 18509, 18538, 18542, 18585, 18603, 18615, 18638, 18638, 18752, 18785, 18790, 18798, 18919, 18970, 18989, 19027, 19079, 19099, 19101, 19109, 19117, 19119, 19137, 19204, 19214, 19225, 19226, 19253, 19283, 19299, 19325, 19396, 19403, 19535, 19618, 19643, 19683, 19699, 19772, 19787, 19844, 19864, 19893, 19893, 19915, 19921, 20012, 20044, 20074, 20145, 20235, 20258, 20260, 20320, 20403, 20412, 20443, 20477, 20482, 20485, 20532, 20559, 20613, 20680, 20785, 20821, 20876, 20929, 21010, 21037, 21043, 21062, 21088, 21151, 21169, 21188, 21226, 21269, 21273, 21281, 21306, 21347, 21364, 21453, 21659, 21692, 21728, 21788, 21837, 21844, 21881, 21961, 22041, 22095, 22202, 22273, 22309, 22338, 22357, 22367, 22398, 22498, 22514, 22562, 22590, 22654, 22921, 23004, 23014, 23025, 23030, 23041, 23125, 23143, 23237, 23297, 23341, 23347, 23368, 23378, 23401, 23435, 23442, 23453, 23506, 23506, 23642, 23685, 23732, 23758, 23759, 23795, 23809, 23931, 23947, 24009, 24068, 24092, 24128, 24180, 24249, 24282, 24463, 24479, 24519, 24565, 24578, 24652, 24712, 24733, 24765, 24843, 24848, 24898, 24905, 24992, 25164, 25178, 25201, 25257, 25265, 25273, 25283, 25459, 25571, 25588, 25661, 25695, 25753, 25775, 25857, 25881, 25933, 25990, 26019, 26156, 26174, 26268, 26293, 26308, 26331, 26417, 26540, 26546, 26605, 26731, 26746, 26804, 26874, 26903, 26992, 27028, 27097, 27098, 27130, 27132, 27155, 27247, 27301, 27350, 27370, 27384, 27393, 27457, 27522, 27569, 27573, 27587, 27626, 27658, 27683, 27690, 27709, 27729, 27733, 27760, 27764, 27813, 27848, 27898, 28019, 28061, 28217, 28274, 28284, 28293, 28331, 28378, 28428, 28497, 28516, 28583, 28586, 28695, 28847, 28942, 28964, 29114, 29153, 29317, 29366, 29408, 29412, 29575, 29609, 29639, 29646, 29801, 30024, 30107, 30125, 30153, 30384, 30404, 30425, 30469, 30533, 30556, 30563, 30672, 30721, 30834, 30847, 30852, 30922, 31106, 31144, 31380, 31387, 31521, 31662, 31777, 31817, 31849, 31937, 32084, 32330, 32387, 32400, 32596, 32606, 32647, 32657, 32785, 32873, 32914, 32943, 32993, 33049, 33117, 33122, 33158, 33166, 33232, 33246, 33305, 33341, 33374, 33386, 33397, 33425, 33426, 33563, 33577, 33605, 33665, 33684, 33689, 33722, 33755, 33816, 33872, 33891, 33914, 33917, 33929, 34065, 34097, 34140, 34181, 34225, 34290, 34355, 34365, 34376, 34478, 34514, 34538, 34576, 34663, 34685, 34735, 34835, 34838, 35032, 35055, 35100, 35190, 35249, 35311, 35321, 35341, 35385, 35423, 35613, 35621, 35637, 35642, 35769, 35772, 35790, 35894, 35896, 35939, 35949, 35952, 35979, 36009, 36113, 36152, 36200, 36240, 36281, 36308, 36411, 36429, 36482, 36507, 36517, 36645, 36649, 36676, 36679, 36699, 36729, 36769, 36781, 36824, 36826, 36838, 36908, 36976, 36983, 37027, 37085, 37148, 37172, 37265, 37273, 37362, 37388, 37479, 37568, 37582, 37586, 37602, 37677, 37682, 37731, 37753, 37781, 37802, 37851, 37911, 37955, 37970, 38034, 38147, 38149, 38184, 38241, 38379, 38433, 38612, 38620, 38667, 38763, 38892, 38909, 39053, 39065, 39071, 39121, 39158, 39217, 39310, 39362, 39401, 39421, 39434, 39454, 39634, 39704, 39825, 39957, 39978, 40018, 40020, 40041, 40138, 40165, 40179, 40197, 40250, 40289, 40305, 40356, 40398, 40401, 40441, 40479, 40545, 40595, 40747, 40765, 40765, 40778, 40800, 40843, 40885, 40981, 41065, 41074, 41143, 41154, 41211, 41265, 41301, 41333, 41378, 41517, 41520, 41568, 41636, 41682, 41706, 41755, 41780, 41792, 41827, 41869, 41946, 41947, 41994, 42133, 42324, 42685, 42733, 42768, 42785, 42834, 42879, 42937, 42957, 42963, 42971, 43109, 43301, 43320, 43349, 43400, 43442, 43445, 43449, 43454, 43655, 43682, 43686, 43697, 43881, 43915, 43933, 43965, 43973, 43986, 44094, 44153, 44240, 44266, 44349, 44527, 44557, 44660, 44673, 44691, 44701, 44863, 44953, 44956, 45044, 45105, 45114, 45178, 45289, 45327, 45347, 45388, 45630, 45743, 45745, 45794, 45897, 46056, 46110, 46195, 46218, 46308, 46320, 46421, 46470, 46491, 46656, 46816, 47014, 47252, 47436, 47465, 47515, 47653, 47690, 47849, 48121, 48163, 48184, 48322, 48450, 48501, 48530, 48650, 48685, 48753, 48759, 48868, 48874, 48908, 48941, 48962, 49057, 49113, 49320, 49504, 49663, 49697, 49701, 49743, 49905, 49910, 49961, 50014, 50018, 50082, 50089, 50112, 50174, 50359, 50385, 50442, 50642, 50739, 50770, 50932, 50957, 50965, 50986, 51101, 51141, 51408, 51607, 52047, 52245, 52312, 52425, 52540, 52628, 52657, 52721, 52725, 52909, 52913, 53066, 53096, 53180, 53186, 53204, 53318, 53318, 53323, 53393, 53438, 53444, 53444, 53469, 53477, 53489, 53627, 53684, 53748, 53783, 54245, 54262, 54369, 54440, 54475, 54601, 54690, 54757, 54816, 54873, 54998, 55059, 55070, 55132, 55162, 55170, 55375, 55461, 55629, 55673, 55778, 55836, 55849, 55854, 55907, 55925, 55958, 56097, 56107, 56114, 56231, 56274, 56314, 56395, 56581, 56593, 56630, 56667, 56681, 56714, 56737, 57014, 57100, 57269, 57416, 57432, 57441, 57463, 57507, 57583, 57628, 57666, 57725, 58048, 58157, 58185, 58290, 58369, 58411, 58482, 58550, 58697, 58913, 58932, 59036, 59098, 59121, 59157, 59229, 59306, 59322, 59380, 59409, 59429, 59468, 59517, 59551, 59560, 59586, 59620, 59645, 59689, 59695, 59740, 59766, 59783, 59870, 59876, 59894, 59917, 60049, 60065, 60084, 60105, 60202, 60228, 60243, 60259, 60314, 60355, 60390, 60451, 60573, 60750, 60814, 60829, 61015, 61323, 61348, 61461, 61484, 61489, 61533, 61658, 61706, 61727, 61742, 61805, 61922, 62010, 62057, 62090, 62127, 62211, 62403, 62486, 62528, 62574, 62730, 62827, 62839, 62888, 63001, 63154, 63214, 63233, 63394, 63436, 63477, 63485, 63525, 63565, 63899, 64104, 64118, 64250, 64290, 64318, 64507, 64650, 64773, 64973, 65284, 65333, 65379, 65379, 65541, 65616, 65689, 65730, 65737, 65792, 65838, 65961, 65990, 66073, 66078, 66129, 66152, 66257, 66286, 66311, 66332, 66378, 66476, 66519, 66655, 66681, 66681, 66697, 66714, 66734, 66748, 66776, 66806, 66829, 66875, 66916, 66922, 66939, 66960, 67105, 67114, 67371, 67417, 67470, 67633, 67676, 67678, 67705, 67709, 67782, 67813, 68061, 68073, 68096, 68129, 68300, 68406, 68492, 68498, 68514, 68586, 68629, 68812, 68845, 68866, 68941, 69033, 69060, 69073, 69113, 69222, 69342, 69345, 69384, 69440, 69536, 69538, 69632, 69641, 69644, 69649, 69729, 69740, 69769, 69773, 69833, 69847, 69849, 69898, 69900, 69969, 70078, 70106, 70140, 70196, 70198, 70306, 70345, 70359, 70378, 70397, 70468, 70516, 70534, 70569, 70636, 70762, 70768, 70798, 70834, 70866, 71004, 71123, 71162, 71233, 71339, 71384, 71429, 71472, 71485, 71505, 71536, 71673, 71674, 71733, 71743, 71795, 71868, 72074, 72212, 72292, 72292, 72346, 72444, 72673, 72691, 72706, 72770, 72977, 73130, 73148, 73189, 73244, 73338, 73361, 73382, 73389, 73422, 73449, 73477, 73567, 73646, 73652, 73675, 73684, 73845, 73873, 73883, 73886, 73901, 73924, 73949, 74065, 74118, 74236, 74284, 74422, 74475, 74522, 74594, 74628, 74632, 74707, 74738, 74889, 75022, 75104, 75125, 75178, 75201, 75222, 75269, 75276, 75357, 75394, 75411, 75497, 75498, 75499, 75515, 75713, 75826, 75864, 75892, 75902, 75922, 75971, 76012, 76030, 76062, 76070, 76113, 76166, 76171, 76200, 76209, 76257, 76498, 76533, 76555, 76560, 76619, 76624, 76694, 76749, 76754, 76773, 76857, 76922, 76987, 77030, 77081, 77107, 77120, 77236, 77340, 77638, 77678, 77837, 77912, 78014, 78109, 78131, 78144, 78152, 78153, 78162, 78206, 78219, 78277, 78290, 78352, 78425, 78475, 78497, 78507, 78509, 78543, 78588, 78597, 78637, 78697, 78711, 78765, 78778, 78782, 78790, 78820, 78837, 78857, 78876, 79005, 79037, 79072, 79185, 79214, 79228, 79265, 79287, 79308, 79381, 79421, 79454, 79517, 79525, 79559, 79573, 79653, 79658, 79820, 79841, 79844, 79886, 80006, 80016, 80047, 80057, 80141, 80150, 80181, 80228, 80229, 80273, 80305, 80366, 80419, 80441, 80487, 80489, 80520, 80541, 80554, 80558, 80574, 80577, 80631, 80781, 80793, 80833, 80861, 80913, 80920, 80954, 81021, 81046, 81065, 81231, 81303, 81338, 81350, 81387, 81397, 81417, 81517, 81550, 81553, 81554, 81610, 81651, 81714, 81796, 81968, 81991, 82026, 82027, 82136, 82187, 82289, 82464, 82553, 82658, 82713, 82741, 82741, 82745, 82746, 82752, 82772, 82809, 82833, 82949, 83005, 83010, 83072, 83085, 83087, 83160, 83171, 83173, 83212, 83235, 83247, 83409, 83466, 83497, 83502, 83538, 83550, 83570, 83617, 83669, 83725, 83853, 83960, 84005, 84045, 84164, 84222, 84241, 84257, 84325, 84341, 84373, 84421, 84442, 84481, 84551, 84581, 84602, 84609, 84685, 84690, 84788, 84809, 84879, 84909, 84938, 84973, 85076, 85081, 85083, 85103, 85119, 85125, 85130, 85197, 85227, 85237, 85307, 85485, 85485, 85490, 85499, 85513, 85536, 85554, 85596, 85598, 85602, 85605, 85639, 85667, 85669, 85693, 85742, 85762, 85776, 85794, 85828, 85914, 86014, 86061, 86075, 86117, 86125, 86133, 86142, 86143, 86146, 86205, 86250, 86271, 86317, 86398, 86405, 86427, 86436, 86460, 86469, 86502, 86561, 86574, 86609, 86687, 86723, 86753, 86759, 86770, 86776, 86778, 86812, 86868, 86925, 86933, 86973, 86975, 87047, 87081, 87172, 87243, 87247, 87330, 87421, 87431, 87447, 87477, 87525, 87714, 87835, 87920, 87973, 88016, 88090, 88125, 88193, 88280, 88293, 88341, 88382, 88406, 88439, 88588, 88598, 88651, 88703, 88806, 88861, 88881, 89025, 89055, 89095, 89113, 89146, 89173, 89226, 89267, 89313, 89316, 89396, 89505, 89519, 89668, 89762, 89802, 89815, 89833, 89845, 89863, 89941, 90058, 90062, 90082, 90135, 90135, 90149, 90222, 90241, 90288, 90365, 90421, 90432, 90525, 90725, 90889, 90895, 90905, 91043, 91071, 91095, 91096, 91133, 91148, 91167, 91174, 91267, 91281, 91291, 91306, 91335, 91368, 91429, 91495, 91542, 91558, 91612, 91645, 91807, 91811, 91852, 91879, 91916, 91925, 91987, 92001, 92124, 92140, 92151, 92186, 92292, 92298, 92309, 92315, 92398, 92409, 92410, 92426, 92481, 92503, 92530, 92565, 92703, 92829, 92834, 92862, 92915, 92933, 92976, 93069, 93116, 93153, 93190, 93218, 93234, 93289, 93383, 93447, 93506, 93628, 93753, 93816, 93825, 93858, 93868, 93878, 94093, 94218, 94224, 94277, 94330, 94398, 94401, 94473, 94573, 94701, 94741, 94864, 94871, 94932, 95055, 95161, 95170, 95195, 95223, 95290, 95290, 95292, 95399, 95500, 95510, 95603, 95608, 95652, 95677, 95679, 95717, 95798, 95871, 95887, 95892, 95951, 95966, 95978, 96070, 96084, 96091, 96123, 96140, 96190, 96417, 96437, 96549, 96558, 96722, 96817, 96836, 96844, 96925, 96946, 96974, 96981, 96981, 96986, 97051, 97068, 97110, 97119, 97141, 97422, 97545, 97626, 97668, 97693, 97767, 97772, 97814, 97885, 98264, 98345, 98407, 98418, 98454, 98457, 98466, 98501, 98510, 98641, 98690, 98809, 98881, 98929, 99048, 99169, 99210, 99272, 99361, 99410, 99418, 99436, 99456, 99538, 99651, 99689, 99758, 99798, 99809, 99826, 99834, 99836, 99913, 99913, 100008, 100074, 100078, 100129, 100214, 100217, 100244, 100267, 100351, 100362, 100386, 100388, 100434, 100454, 100516, 100616, 100681, 100708, 100711, 100719, 100825, 100846, 100858, 100885, 100967, 100997, 101049, 101070, 101158, 101246, 101355, 101373, 101493, 101523, 101595, 101614, 101665, 101722, 101723, 101756, 101789, 101824, 101841, 101892, 101914, 101944, 101963, 102015, 102025, 102214, 102233, 102273, 102353, 102365, 102378, 102476, 102478, 102492, 102494, 102501, 102545, 102611, 102617, 102727, 102756, 102765, 102806, 102810, 102925, 102929, 103017, 103033, 103069, 103190, 103197, 103221, 103229, 103313, 103372, 103385, 103432, 103506, 103539, 103566, 103608, 103652, 103653, 103733, 103768, 103841, 103949, 104029, 104030, 104038, 104082, 104158, 104234, 104287, 104306, 104333, 104347, 104610, 104634, 104760, 104782, 104862, 104873, 104894, 104941, 105037, 105089, 105208, 105362, 105402, 105450, 105481, 105511, 105549, 105563, 105644, 105722, 105736, 105753, 105781, 105807, 105970, 106058, 106072, 106178, 106179, 106210, 106241, 106267, 106296, 106330, 106331, 106508, 106527, 106724, 106735, 106745, 106784, 106883, 107038, 107339, 107466, 107469, 107561, 107717, 107738, 107773, 107774, 107776, 107880, 107899, 107901, 107989, 108014, 108089, 108175, 108194, 108197, 108197, 108338, 108470, 108650, 108741, 108799, 108821, 108826, 108876, 108904, 108983, 108986, 109154, 109158, 109255, 109324, 109353, 109462, 109525, 109602, 109854, 109869, 109899, 110120, 110218, 110280, 110457, 110644, 110743, 110873, 110881, 110949, 110972, 111092, 111136, 111146, 111198, 111239, 111266, 111331, 111501, 111575, 111589, 111689, 111727, 111822, 111904, 112094, 112173, 112257, 112309, 112380, 112533, 112561, 112568, 112586, 112629, 112654, 112767, 112854, 113306, 113505, 113521, 113533, 113566, 113623, 113654, 113718, 113981, 114013, 114054, 114110, 114170, 114226, 114379, 114456, 114517, 114560, 114797, 114878, 114900, 114918, 115018, 115098, 115127, 115154, 115181, 115225, 115234, 115254, 115287, 115422, 115465, 115472, 115516, 115574, 115705, 115831, 115883, 115938, 115945, 115951, 116024, 116070, 116254, 116298, 116397, 116494, 116584, 116609, 116683, 116685, 116718, 116874, 116928, 116962, 117010, 117058, 117089, 117297, 117354, 117366, 117392, 117425, 117474, 117511, 117538, 117583, 117663, 117682, 117783, 117797, 117813, 117871, 117909, 117910, 117930, 117961, 118002, 118018, 118081, 118082, 118090, 118112, 118209, 118261, 118261, 118314, 118385, 118561, 118565, 118631, 118669, 118705, 118733, 118744, 118835, 118877, 118907, 118927, 119089, 119101, 119123, 119141, 119142, 119146, 119289, 119399, 119405, 119482, 119482, 119527, 119553, 119607, 119632, 119668, 119669, 119794, 119822, 119831, 119870, 119879, 119880, 119884, 119919, 119952, 120022, 120029, 120052, 120122, 120151, 120231, 120245, 120316, 120400, 120577, 120593, 120640, 120643, 120722, 120857, 120882, 120913, 121037, 121038, 121044, 121112, 121113, 121116, 121128, 121156, 121196, 121227, 121280, 121305, 121362, 121393, 121469, 121477, 121613, 121625, 121797, 121820, 121831, 121877, 121895, 121896, 121916, 121932, 122045, 122114, 122144, 122189, 122281, 122389, 122475, 122565, 122811, 122822, 122870, 122949, 123084, 123161, 123172, 123250, 123251, 123270, 123327, 123334, 123422, 123454, 123490, 123496, 123596, 123647, 123725, 123995, 124026, 124088, 124167, 124205, 124288, 124377, 124435, 124527, 124570, 124581, 124618, 124620, 124622, 124634, 124640, 124673, 124853, 124870, 124989, 125001, 125044, 125178, 125190, 125213, 125228, 125367, 125373, 125497, 125535, 125560, 125578, 125687, 125695, 125696, 125725, 125762, 125775, 125786, 125821, 125854, 125898, 125946, 125998, 126011, 126019, 126194, 126243, 126251, 126270, 126517, 126557, 126597, 126679, 126793, 126799, 126847, 126972, 126999, 127009, 127059, 127102, 127311, 127366, 127405, 127409, 127538, 127645, 127717, 127726, 127737, 127809, 127892, 127952, 127987, 128005, 128094, 128239, 128261, 128290, 128301, 128405, 128470, 128591, 128666, 128751, 128781, 128878, 128888, 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