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Totient Answers For The First 1000 Integers

The table below lists all totient (φ) answers to the first 1000 integers. We can be certain we're listing all since it is a mathematical fact that x/φ(x) is never more than the square root of x, (with only two exceptions, 2 and 6).

You might notice that in the table below, 1 is the only odd number in the n column. 1 is a special case. It is the only number that is its own φ value and the only odd number to ever be a φ value. So we need not list any other odd n, since the answer will always be the same: they're nontotients. There are even numbers that are never a φ value either. I don't know if there is a pattern to them. They are listed below as "Nontotients".

You might also notice that some numbers have more φ answers than others. A number that has more φ answers than any number before it is called a highly totient number. The first few highly totient numbers are 1, 2, 4, 8, 12, 24, 48, 72, 144, 240, 432, 480, 576, 720, as you can verify from the table.

nx such that φ(x) = n
11, 2
23, 4, 6
45, 8, 10, 12
67, 9, 14, 18
815, 16, 20, 24, 30
1011, 22
1011, 22
1213, 21, 26, 28, 36, 42
14Nontotient
1617, 32, 34, 40, 48, 60
1819, 27, 38, 54
2025, 33, 44, 50, 66
2223, 46
2435, 39, 45, 52, 56, 70, 72, 78, 84, 90
26Nontotient
2829, 58
3031, 62
3251, 64, 68, 80, 96, 102, 120
34Nontotient
3637, 57, 63, 74, 76, 108, 114, 126
38Nontotient
4041, 55, 75, 82, 88, 100, 110, 132, 150
4243, 49, 86, 98
4469, 92, 138
4647, 94
4865, 104, 105, 112, 130, 140, 144, 156, 168, 180, 210
50Nontotient
5253, 106
5481, 162
5687, 116, 174
5859, 118
6061, 77, 93, 99, 122, 124, 154, 186, 198
62Nontotient
6485, 128, 136, 160, 170, 192, 204, 240
6667, 134
68Nontotient
7071, 142
7273, 91, 95, 111, 117, 135, 146, 148, 152, 182, 190, 216, 222, 228, 234, 252, 270
74Nontotient
76Nontotient
7879, 158
80123, 164, 165, 176, 200, 220, 246, 264, 300, 330
8283, 166
84129, 147, 172, 196, 258, 294
86Nontotient
8889, 115, 178, 184, 230, 276
90Nontotient
92141, 188, 282
94Nontotient
9697, 119, 153, 194, 195, 208, 224, 238, 260, 280, 288, 306, 312, 336, 360, 390, 420
98Nontotient
100101, 125, 202, 250
102103, 206
104159, 212, 318
106107, 214
108109, 133, 171, 189, 218, 266, 324, 342, 378
110121, 242
112113, 145, 226, 232, 290, 348
114Nontotient
116177, 236, 354
118Nontotient
120143, 155, 175, 183, 225, 231, 244, 248, 286, 308, 310, 350, 366, 372, 396, 450, 462
122Nontotient
124Nontotient
126127, 254
128255, 256, 272, 320, 340, 384, 408, 480, 510
130131, 262
132161, 201, 207, 268, 322, 402, 414
134Nontotient
136137, 274
138139, 278
140213, 284, 426
142Nontotient
144185, 219, 273, 285, 292, 296, 304, 315, 364, 370, 380, 432, 438, 444, 456, 468, 504, 540, 546, 570, 630
146Nontotient
148149, 298
150151, 302
152Nontotient
154Nontotient
156157, 169, 237, 314, 316, 338, 474
158Nontotient
160187, 205, 328, 352, 374, 400, 410, 440, 492, 528, 600, 660
162163, 243, 326, 486
164249, 332, 498
166167, 334
168203, 215, 245, 261, 344, 392, 406, 430, 490, 516, 522, 588
170Nontotient
172173, 346
174Nontotient
176267, 345, 356, 368, 460, 534, 552, 690
178179, 358
180181, 209, 217, 279, 297, 362, 418, 434, 558, 594
182Nontotient
184235, 376, 470, 564
186Nontotient
188Nontotient
190191, 382
192193, 221, 291, 357, 386, 388, 416, 442, 448, 476, 520, 560, 576, 582, 612, 624, 672, 714, 720, 780, 840
194Nontotient
196197, 394
198199, 398
200275, 303, 375, 404, 500, 550, 606, 750
202Nontotient
204309, 412, 618
206Nontotient
208265, 424, 530, 636
210211, 422
212321, 428, 642
214Nontotient
216247, 259, 327, 333, 351, 399, 405, 436, 494, 518, 532, 648, 654, 666, 684, 702, 756, 798, 810
218Nontotient
220253, 363, 484, 506, 726
222223, 446
224339, 435, 452, 464, 580, 678, 696, 870
226227, 454
228229, 458
230Nontotient
232233, 295, 466, 472, 590, 708
234Nontotient
236Nontotient
238239, 478
240241, 287, 305, 325, 369, 385, 429, 465, 482, 488, 495, 496, 525, 572, 574, 610, 616, 620, 650, 700, 732, 738, 744, 770, 792, 858, 900, 924, 930, 990, 1050
242Nontotient
244Nontotient
246Nontotient
248Nontotient
250251, 502
252301, 381, 387, 441, 508, 602, 762, 774, 882
254Nontotient
256257, 512, 514, 544, 640, 680, 768, 816, 960, 1020
258Nontotient
260393, 524, 786
262263, 526
264299, 335, 483, 536, 598, 644, 670, 804, 828, 966
266Nontotient
268269, 538
270271, 542
272289, 411, 548, 578, 822
274Nontotient
276277, 329, 417, 423, 554, 556, 658, 834, 846
278Nontotient
280281, 319, 355, 562, 568, 638, 710, 852
282283, 566
284Nontotient
286Nontotient
288323, 365, 455, 459, 555, 584, 585, 592, 608, 646, 728, 730, 740, 760, 864, 876, 888, 910, 912, 918, 936, 1008, 1080, 1092, 1110, 1140, 1170, 1260
290Nontotient
292293, 586
294343, 686
296447, 596, 894
298Nontotient
300341, 453, 604, 682, 906
302Nontotient
304Nontotient
306307, 614
308Nontotient
310311, 622
312313, 371, 395, 471, 477, 507, 626, 628, 632, 676, 742, 790, 942, 948, 954, 1014
314Nontotient
316317, 634
318Nontotient
320425, 561, 615, 656, 704, 748, 800, 820, 850, 880, 984, 1056, 1122, 1200, 1230, 1320
322Nontotient
324489, 513, 567, 652, 972, 978, 1026, 1134
326Nontotient
328415, 664, 830, 996
330331, 662
332501, 668, 1002
334Nontotient
336337, 377, 609, 645, 674, 688, 735, 754, 784, 812, 860, 980, 1032, 1044, 1176, 1218, 1290, 1470
338Nontotient
340Nontotient
342361, 722
344519, 692, 1038
346347, 694
348349, 413, 531, 698, 826, 1062
350Nontotient
352353, 391, 445, 706, 712, 736, 782, 890, 920, 1068, 1104, 1380
354Nontotient
356537, 716, 1074
358359, 718
360403, 407, 427, 475, 543, 549, 627, 651, 675, 693, 724, 806, 814, 836, 854, 868, 950, 1086, 1098, 1116, 1188, 1254, 1302, 1350, 1386
362Nontotient
364Nontotient
366367, 734
368705, 752, 940, 1128, 1410
370Nontotient
372373, 746
374Nontotient
376Nontotient
378379, 758
380573, 764, 1146
382383, 766
384485, 579, 595, 663, 765, 772, 776, 832, 884, 896, 952, 970, 1040, 1120, 1152, 1158, 1164, 1190, 1224, 1248, 1326, 1344, 1428, 1440, 1530, 1560, 1680
386Nontotient
388389, 778
390Nontotient
392591, 788, 1182
394Nontotient
396397, 437, 469, 597, 603, 621, 794, 796, 874, 938, 1194, 1206, 1242
398Nontotient
400401, 451, 505, 802, 808, 825, 902, 1000, 1010, 1100, 1212, 1500, 1650
402Nontotient
404Nontotient
406Nontotient
408409, 515, 818, 824, 1030, 1236
410Nontotient
412Nontotient
414Nontotient
416795, 848, 1060, 1272, 1590
418419, 838
420421, 473, 497, 539, 633, 639, 842, 844, 946, 994, 1078, 1266, 1278
422Nontotient
424535, 856, 1070, 1284
426Nontotient
428Nontotient
430431, 862
432433, 481, 511, 545, 657, 665, 741, 777, 819, 855, 866, 872, 945, 962, 988, 1022, 1036, 1064, 1090, 1296, 1308, 1314, 1330, 1332, 1368, 1404, 1482, 1512, 1554, 1596, 1620, 1638, 1710, 1890
434Nontotient
436Nontotient
438439, 878
440575, 605, 759, 968, 1012, 1150, 1210, 1452, 1518
442443, 886
444669, 892, 1338
446Nontotient
448449, 493, 565, 898, 904, 928, 986, 1130, 1160, 1356, 1392, 1740
450Nontotient
452681, 908, 1362
454Nontotient
456457, 687, 914, 916, 1374
458Nontotient
460461, 517, 922, 1034
462463, 926
464699, 885, 932, 944, 1180, 1398, 1416, 1770
466467, 934
468553, 711, 1106, 1422
470Nontotient
472Nontotient
474Nontotient
476717, 956, 1434
478479, 958
480527, 533, 715, 723, 861, 915, 964, 975, 976, 992, 1054, 1066, 1144, 1148, 1155, 1220, 1232, 1240, 1300, 1400, 1430, 1446, 1464, 1476, 1488, 1540, 1584, 1716, 1722, 1800, 1830, 1848, 1860, 1950, 1980, 2100, 2310
482Nontotient
484Nontotient
486487, 729, 974, 1458
488Nontotient
490491, 982
492581, 747, 1162, 1494
494Nontotient
496Nontotient
498499, 998
500625, 753, 1004, 1250, 1506
500625, 753, 1004, 1250, 1506
502503, 1006
504551, 559, 635, 637, 783, 903, 1016, 1102, 1118, 1204, 1270, 1274, 1524, 1548, 1566, 1764, 1806
506529, 1058
508509, 1018
510Nontotient
512771, 1024, 1028, 1088, 1280, 1360, 1536, 1542, 1632, 1920, 2040
514Nontotient
516Nontotient
518Nontotient
520521, 583, 655, 1042, 1048, 1166, 1310, 1572
522523, 1046
524789, 1052, 1578
526Nontotient
528623, 801, 805, 897, 1005, 1035, 1072, 1196, 1246, 1288, 1340, 1602, 1608, 1610, 1656, 1794, 1932, 2010, 2070
530Nontotient
532Nontotient
534Nontotient
536807, 1076, 1614
538Nontotient
540541, 589, 813, 837, 891, 1082, 1084, 1178, 1626, 1674, 1782
542Nontotient
544685, 867, 1096, 1156, 1370, 1644, 1734
546547, 1094
548Nontotient
550Nontotient
552611, 695, 831, 987, 1108, 1112, 1222, 1316, 1390, 1662, 1668, 1692, 1974
554Nontotient
556557, 1114
558Nontotient
560725, 843, 957, 1065, 1124, 1136, 1276, 1420, 1450, 1686, 1704, 1914, 2130
562563, 1126
564849, 1132, 1698
566Nontotient
568569, 1138
570571, 1142
572Nontotient
574Nontotient
576577, 629, 679, 873, 969, 1071, 1095, 1154, 1168, 1184, 1216, 1258, 1292, 1358, 1365, 1456, 1460, 1480, 1520, 1728, 1746, 1752, 1776, 1820, 1824, 1836, 1872, 1938, 2016, 2142, 2160, 2184, 2190, 2220, 2280, 2340, 2520, 2730
578Nontotient
580649, 1298
582Nontotient
584879, 1172, 1758
586587, 1174
5881029, 1372, 2058
590Nontotient
592593, 745, 1186, 1192, 1490, 1788
594Nontotient
596Nontotient
598599, 1198
600601, 671, 707, 755, 775, 875, 909, 1023, 1125, 1202, 1208, 1342, 1364, 1414, 1510, 1550, 1750, 1812, 1818, 2046, 2250
602Nontotient
604Nontotient
606607, 1214
608Nontotient
610Nontotient
612613, 721, 921, 927, 1226, 1228, 1442, 1842, 1854
614Nontotient
616617, 667, 1234, 1334
618619, 1238
620933, 1244, 1866
622Nontotient
624689, 785, 845, 939, 1113, 1185, 1252, 1256, 1264, 1352, 1378, 1484, 1570, 1580, 1690, 1878, 1884, 1896, 1908, 2028, 2226, 2370
626Nontotient
628Nontotient
630631, 1262
632951, 1268, 1902
634Nontotient
636749, 963, 1498, 1926
638Nontotient
640641, 697, 935, 1275, 1282, 1312, 1394, 1408, 1496, 1600, 1640, 1700, 1760, 1870, 1968, 2112, 2244, 2400, 2460, 2550, 2640
642643, 1286
644Nontotient
646647, 1294
648703, 763, 815, 981, 999, 1053, 1197, 1215, 1304, 1406, 1526, 1630, 1944, 1956, 1962, 1998, 2052, 2106, 2268, 2394, 2430
650Nontotient
652653, 1306
654Nontotient
6561245, 1328, 1660, 1992, 2490
658659, 1318
660661, 713, 737, 847, 993, 1089, 1322, 1324, 1426, 1474, 1694, 1986, 2178
662Nontotient
664835, 1336, 1670, 2004
666Nontotient
668Nontotient
670Nontotient
672673, 731, 791, 833, 1011, 1015, 1017, 1131, 1305, 1346, 1348, 1376, 1462, 1508, 1568, 1582, 1624, 1666, 1720, 1960, 2022, 2030, 2034, 2064, 2088, 2262, 2352, 2436, 2580, 2610, 2940
674Nontotient
676677, 1354
678Nontotient
680Nontotient
682683, 1366
6841083, 1444, 2166
686Nontotient
688865, 1384, 1730, 2076
690691, 1382
6921041, 1388, 2082
694Nontotient
696767, 1047, 1239, 1396, 1534, 1652, 2094, 2124, 2478
698Nontotient
700701, 781, 1402, 1562
702Nontotient
7041059, 1173, 1335, 1412, 1424, 1472, 1564, 1780, 1840, 2118, 2136, 2208, 2346, 2670, 2760
706Nontotient
708709, 1418
710Nontotient
712895, 1432, 1790, 2148
714Nontotient
7161077, 1436, 2154
718719, 1438
720779, 793, 803, 905, 925, 1001, 1045, 1085, 1107, 1209, 1221, 1281, 1287, 1395, 1425, 1448, 1485, 1558, 1575, 1586, 1606, 1612, 1628, 1672, 1708, 1736, 1810, 1850, 1900, 2002, 2090, 2170, 2172, 2196, 2214, 2232, 2376, 2418, 2442, 2508, 2562, 2574, 2604, 2700, 2772, 2790, 2850, 2970, 3150
722Nontotient
724Nontotient
726727, 1454
728Nontotient
730Nontotient
732733, 1101, 1466, 1468, 2202
734Nontotient
736799, 1504, 1598, 1880, 2256, 2820
738739, 1478
740Nontotient
742743, 1486
7441119, 1492, 2238
746Nontotient
748Nontotient
750751, 1502
752Nontotient
754Nontotient
756757, 817, 889, 931, 1137, 1143, 1161, 1323, 1514, 1516, 1634, 1778, 1862, 2274, 2286, 2322, 2646
758Nontotient
760761, 955, 1522, 1528, 1910, 2292
762Nontotient
7641149, 1532, 2298
766Nontotient
768769, 965, 1105, 1455, 1538, 1544, 1552, 1664, 1768, 1785, 1792, 1904, 1930, 1940, 2080, 2210, 2240, 2304, 2316, 2328, 2380, 2448, 2496, 2652, 2688, 2856, 2880, 2910, 3060, 3120, 3360, 3570
770Nontotient
772773, 1546
774Nontotient
7761167, 1556, 2334
778Nontotient
780869, 917, 1179, 1738, 1834, 2358
782Nontotient
784985, 1576, 1970, 2364
786787, 1574
788Nontotient
790Nontotient
792851, 871, 995, 1191, 1311, 1407, 1449, 1588, 1592, 1702, 1742, 1748, 1876, 1990, 2382, 2388, 2412, 2484, 2622, 2814, 2898
794Nontotient
796797, 1594
798Nontotient
8001025, 1203, 1353, 1515, 1604, 1616, 1804, 2000, 2020, 2050, 2200, 2406, 2424, 2706, 3000, 3030, 3300
802Nontotient
804Nontotient
806Nontotient
808809, 1618
810811, 1622
812841, 1682
814Nontotient
816959, 1227, 1233, 1545, 1636, 1648, 1918, 2060, 2454, 2466, 2472, 3090
818Nontotient
820821, 913, 1642, 1826
822823, 1646
824Nontotient
826827, 1654
828829, 893, 973, 1251, 1269, 1658, 1786, 1946, 2502, 2538
830Nontotient
832901, 1696, 1802, 2120, 2544, 3180
834Nontotient
8361257, 1676, 2514
838839, 1678
840899, 923, 1055, 1075, 1225, 1263, 1419, 1491, 1617, 1684, 1688, 1798, 1846, 1892, 1988, 2110, 2150, 2156, 2450, 2526, 2532, 2556, 2838, 2982, 3234
842Nontotient
844Nontotient
846Nontotient
8481605, 1712, 2140, 2568, 3210
850Nontotient
852853, 1706
854Nontotient
856857, 1714
858859, 1718
8601293, 1724, 2586
862863, 1726
864949, 1235, 1295, 1299, 1377, 1443, 1533, 1635, 1665, 1732, 1744, 1755, 1898, 1924, 1976, 1995, 2044, 2072, 2128, 2180, 2470, 2590, 2592, 2598, 2616, 2628, 2660, 2664, 2736, 2754, 2808, 2886, 2964, 3024, 3066, 3108, 3192, 3240, 3270, 3276, 3330, 3420, 3510, 3780, 3990
866Nontotient
868Nontotient
870Nontotient
872Nontotient
874
876877, 1317, 1754, 1756, 2634
878Nontotient
880881, 943, 979, 1265, 1725, 1762, 1815, 1886, 1936, 1958, 2024, 2300, 2420, 2530, 2904, 3036, 3450, 3630
882883, 1766
8841329, 1772, 2658
886887, 1774
8881043, 1115, 1341, 1784, 2086, 2230, 2676, 2682
890Nontotient
892Nontotient
894Nontotient
8961347, 1479, 1695, 1796, 1808, 1856, 1972, 2260, 2320, 2694, 2712, 2784, 2958, 3390, 3480
898Nontotient
9001057, 1359, 2114, 2718
902Nontotient
9041135, 1816, 2270, 2724
906907, 1814
908Nontotient
910911, 1822
9121145, 1371, 1828, 1832, 2290, 2742, 2748
914Nontotient
916Nontotient
918919, 1838
9201175, 1383, 1551, 1844, 2068, 2350, 2766, 3102
922Nontotient
924989, 1127, 1389, 1852, 1978, 2254, 2778
926Nontotient
928929, 1003, 1165, 1858, 1864, 1888, 2006, 2330, 2360, 2796, 2832, 3540
930961, 1922
9321401, 1868, 2802
934Nontotient
936937, 1007, 1027, 1099, 1183, 1413, 1431, 1521, 1659, 1874, 2014, 2054, 2198, 2212, 2366, 2826, 2844, 2862, 3042, 3318
938Nontotient
940941, 1882
942Nontotient
944Nontotient
946947, 1894
948Nontotient
950
952953, 1195, 1906, 1912, 2390, 2868
954Nontotient
9561437, 1916, 2874
958Nontotient
9601037, 1067, 1205, 1309, 1435, 1581, 1599, 1683, 1845, 1928, 1952, 1984, 2074, 2108, 2132, 2134, 2145, 2288, 2296, 2410, 2440, 2464, 2480, 2600, 2618, 2800, 2860, 2870, 2892, 2928, 2952, 2976, 3080, 3162, 3168, 3198, 3366, 3432, 3444, 3600, 3660, 3690, 3696, 3720, 3900, 3960, 4200, 4290, 4620
962Nontotient
964Nontotient
966967, 1934
968Nontotient
970971, 1942
9721141, 1461, 1467, 1539, 1701, 1948, 2282, 2916, 2922, 2934, 3078, 3402
974Nontotient
976977, 1954
978Nontotient
9801473, 1964, 2946
982983, 1966
9841079, 1743, 2158, 2324, 2988, 3486
986Nontotient
988Nontotient
990991, 1982
992Nontotient
994Nontotient
996997, 1169, 1497, 1503, 1994, 1996, 2338, 2994, 3006
998Nontotient
10001111, 1255, 1375, 1875, 2008, 2222, 2500, 2510, 2750, 3012, 3750

Values calculated with Mathematica 4, with the command Table[{n, Select[Range[n + 1, n^2 + 2], EulerPhi[#] == n &]}, {n, 2, 1000, 2}]. As noted above, 1 is a special case.