A number n for which the number of digits in its prime factorization is equal to the number of digits of n. more
The first 600 equidigital numbers :
1,
2,
3,
5,
7,
10,
11,
13,
14,
15,
16,
17,
19,
21,
23,
25,
27,
29,
31,
32,
35,
37,
41,
43,
47,
49,
53,
59,
61,
64,
67,
71,
73,
79,
81,
83,
89,
97,
101,
103,
105,
106,
107,
109,
111,
112,
113,
115,
118,
119,
121,
122,
123,
127,
129,
131,
133,
134,
135,
137,
139,
141,
142,
145,
146,
147,
149,
151,
155,
157,
158,
159,
160,
161,
162,
163,
166,
167,
169,
173,
175,
177,
178,
179,
181,
183,
185,
189,
191,
192,
193,
194,
197,
199,
201,
203,
205,
211,
213,
215,
217,
219,
223,
224,
227,
229,
233,
235,
237,
239,
241,
245,
249,
250,
251,
257,
259,
263,
265,
267,
269,
271,
277,
281,
283,
287,
289,
291,
293,
295,
301,
305,
307,
311,
313,
317,
320,
329,
331,
335,
337,
347,
349,
353,
355,
359,
361,
365,
367,
371,
373,
375,
379,
383,
384,
389,
395,
397,
401,
405,
409,
413,
415,
419,
421,
427,
431,
433,
439,
443,
445,
448,
449,
457,
461,
463,
467,
469,
479,
485,
486,
487,
491,
497,
499,
503,
509,
511,
521,
523,
529,
541,
547,
553,
557,
563,
567,
569,
571,
577,
581,
587,
593,
599,
601,
607,
613,
617,
619,
623,
631,
640,
641,
643,
647,
653,
659,
661,
673,
677,
679,
683,
686,
691,
701,
709,
719,
727,
733,
739,
743,
751,
757,
761,
768,
769,
773,
787,
797,
809,
811,
821,
823,
827,
829,
839,
841,
853,
857,
859,
863,
875,
877,
881,
883,
887,
896,
907,
911,
919,
929,
937,
941,
947,
953,
961,
967,
971,
977,
983,
991,
997,
1000,
1003,
1005,
1006,
1007,
1009,
1011,
1013,
1015,
1018,
1019,
1021,
1022,
1025,
1027,
1031,
1033,
1037,
1039,
1041,
1042,
1043,
1046,
1047,
1049,
1051,
1053,
1055,
1057,
1058,
1059,
1061,
1063,
1065,
1067,
1069,
1072,
1073,
1075,
1077,
1079,
1081,
1082,
1083,
1085,
1087,
1088,
1091,
1093,
1094,
1095,
1097,
1099,
1101,
1103,
1106,
1107,
1109,
1113,
1114,
1115,
1117,
1119,
1120,
1121,
1123,
1125,
1126,
1127,
1129,
1134,
1135,
1136,
1137,
1138,
1139,
1141,
1142,
1145,
1147,
1149,
1151,
1152,
1153,
1154,
1157,
1159,
1161,
1162,
1163,
1165,
1167,
1168,
1169,
1171,
1174,
1175,
1181,
1183,
1184,
1185,
1186,
1187,
1189,
1191,
1193,
1195,
1198,
1201,
1202,
1203,
1205,
1207,
1211,
1213,
1214,
1216,
1217,
1219,
1223,
1225,
1226,
1227,
1229,
1231,
1234,
1237,
1238,
1239,
1241,
1245,
1246,
1247,
1249,
1253,
1255,
1257,
1259,
1261,
1262,
1263,
1264,
1267,
1269,
1271,
1273,
1277,
1279,
1281,
1282,
1283,
1285,
1286,
1289,
1291,
1293,
1294,
1295,
1296,
1297,
1299,
1301,
1303,
1306,
1307,
1312,
1315,
1317,
1318,
1319,
1321,
1322,
1323,
1325,
1327,
1328,
1329,
1333,
1335,
1337,
1343,
1344,
1345,
1346,
1347,
1349,
1351,
1354,
1355,
1357,
1358,
1361,
1363,
1366,
1367,
1371,
1372,
1373,
1375,
1376,
1377,
1379,
1381,
1382,
1383,
1385,
1387,
1389,
1393,
1399,
1401,
1402,
1403,
1405,
1407,
1408,
1409,
1411,
1415,
1418,
1421,
1423,
1424,
1427,
1429,
1431,
1433,
1435,
1437,
1438,
1439,
1445,
1447,
1451,
1453,
1454,
1455,
1457,
1459,
1461,
1465,
1466,
1471,
1472,
1473,
1475,
1477,
1478,
1481,
1483,
1486,
1487,
1489,
1491,
1493,
1497,
1499,
1501,
1502,
1504,
1505,
1509,
1511,
1513,
1514,
1517,
1519,
1522,
1523,
1525,
1527,
1531,
1533,
1535,
1537,
1538,
1539,
1541,
1543,
1546,
1549,
1552,
1553,
1555,
1559,
1561,
1563,
1565,
1567,
1568,
1569,
1571,
1574,
1577,
1579,
1583,
1585,
1587,
1589,
1591,
1593,
1594,
1597,
1600,
1601,
1603,
1607,
1609,
1613,
1618,
1619,
1621,
1622,
1623,
1625,
1627,
1631,
1633,
1637,
1641,
1642,
1643,
1645,
1646,
1647,
1649,
1654,
1655.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 4624231 values, from 1 to 19999999).
n\r | 0 | 1 |
2 | 727171 | 3897060 | 2 |
3 | 648693 | 1987777 | 1987761 | 3 |
4 | 240890 | 1948415 | 486281 | 1948645 | 4 |
5 | 558894 | 1016239 | 1016456 | 1016440 | 1016202 | 5 |
6 | 55120 | 1651556 | 335830 | 593573 | 336221 | 1651931 | 6 |
7 | 536873 | 681059 | 681231 | 681451 | 681244 | 681184 | 681189 | 7 |
8 | 234801 | 974134 | 243085 | 974237 | 6089 | 974281 | 243196 | 974408 | 8 |
9 | 147508 | 662757 | 662671 | 250464 | 662503 | 662426 | 250721 | 662517 | 662664 | 9 |
10 | 47042 | 846280 | 170083 | 846416 | 170063 | 511852 | 169959 | 846373 | 170024 | 846139 | 10 |
11 | 25007 | 459785 | 460002 | 459999 | 459789 | 459793 | 460079 | 460059 | 460179 | 459681 | 459858 |
A pictorial representation of the table above
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.