31:
605:
897:
1511:
1779:
1558:; a mesh of progressively finer sizes allows for arbitrary accuracy. For high precision calculations (hundreds of digits), a recursive series expansion about the midpoints of the intervals is superior.
587:
337:
612:
The main purpose of the
Dickmanâde Bruijn function is to estimate the frequency of smooth numbers at a given size. This can be used to optimize various number-theoretical algorithms such as
195:
795:
731:
1274:
683:
957:
1009:
1364:
803:
452:
46:, and the vertical axis is the value of the function. The graph nearly makes a downward line on the logarithmic scale, demonstrating that the logarithm of the function is
1825:
1669:
1599:
1376:
1313:
375:
233:
1549:
1223:
484:
1628:
1063:
120:
1986:
1033:
265:
2043:
1851:
1879:
81:, who defined it in his only mathematical publication, which is not easily available, and later studied by the Dutch mathematician
1685:
2309:
495:
1967:
1830:
737:
273:
127:
608:
The
Dickmanâde Bruijn used to calculate the probability that the largest and 2nd largest factor of x is less than x^a
2314:
2081:
1888:
Discussion: an unsuccessful search for a source of
Dickman's paper, and suggestions on several others on the topic.
82:
751:
135:
1849:
Dickman, K. (1930). "On the frequency of numbers containing prime factors of a certain relative magnitude".
691:
1228:
625:
892:{\displaystyle \rho (u)\sim {\frac {1}{\xi {\sqrt {2\pi u}}}}\cdot \exp(-u\xi +\operatorname {Ei} (\xi ))}
55:
916:
965:
47:
1322:
407:
2126:
1880:"nt.number theory - Reference request: Dickman, On the frequency of numbers containing prime factors"
1860:
903:
1506:{\displaystyle \rho (u)=1-(1-\log(u-1))\log(u)+\operatorname {Li} _{2}(1-u)+{\frac {\pi ^{2}}{12}}.}
2077:"On the Numerical Solution of a Differential-Difference Equation Arising in Analytic Number Theory"
123:
1800:
1633:
1569:
78:
2269:
2243:
2218:
1283:
345:
203:
1527:
1201:
457:
2282:
2234:
1933:
1899:
1790:
1604:
1196:
1039:
96:
2035:
2253:
2196:
2167:
2134:
2090:
2052:
2006:
1555:
70:
30:
2265:
2020:
2261:
2016:
604:
17:
2232:
Soundararajan, Kannan (2012). "An asymptotic expansion related to the
Dickman function".
2130:
1864:
1971:
1018:
593:
390:
250:
2172:
2155:
2095:
2076:
2303:
2285:
2273:
1521:
382:
74:
2011:
1794:
2139:
613:
487:
2200:
2111:
2257:
2290:
2187:
Friedlander, John B. (1976). "Integers free from large and small primes".
2217:
Broadhurst, David (2010). "Dickman polylogarithms and their constants".
2156:"Numerical Solution of Some Classical Differential-Difference Equations"
740:
has an alternate definition in terms of the
Dickmanâde Bruijn function.
42:) plotted on a logarithmic scale. The horizontal axis is the argument
2057:
2248:
2223:
603:
29:
1554:
An alternate method is computing lower and upper bounds with the
2154:
Marsaglia, George; Zaman, Arif; Marsaglia, John C. W. (1989).
1774:{\displaystyle \Psi (x,x^{1/a},x^{1/b})\sim x\sigma (b,a).\,}
1675:-smooth integers with at most one prime factor greater than
1793:, a function used similarly to estimate the number of
582:{\displaystyle \Psi (x,x^{1/a})=x\rho (a)+O(x/\log x)}
1974:
1803:
1688:
1636:
1607:
1572:
1530:
1379:
1325:
1286:
1231:
1204:
1042:
1021:
968:
919:
806:
754:
694:
628:
498:
460:
410:
400:
Ramaswami later gave a rigorous proof that for fixed
348:
276:
253:
206:
138:
99:
77:
up to a given bound. It was first studied by actuary
1671:similar to de Bruijn's, but counting the number of
1980:
1819:
1773:
1663:
1622:
1593:
1543:
1505:
1358:
1307:
1268:
1217:
1057:
1027:
1003:
951:
891:
789:
725:
677:
581:
478:
446:
369:
331:
259:
227:
189:
114:
2070:
2068:
332:{\displaystyle \Psi (x,x^{1/a})\sim x\rho (a)\,}
1630:. This function is used to estimate a function
1968:"On the number of positive integers less than
1999:Bulletin of the American Mathematical Society
1988:and free of prime divisors greater than
1566:Friedlander defines a two-dimensional analog
8:
2044:Journal de théorie des nombres de Bordeaux
2247:
2222:
2171:
2138:
2112:"Asymptotic Semismoothness Probabilities"
2094:
2056:
2010:
1973:
1938:and free of prime factors >
1808:
1802:
1770:
1730:
1726:
1709:
1705:
1687:
1635:
1606:
1571:
1551:can be calculated using infinite series.
1535:
1529:
1489:
1483:
1456:
1378:
1324:
1285:
1236:
1230:
1209:
1203:
1041:
1020:
987:
967:
948:
924:
918:
831:
822:
805:
790:{\displaystyle \rho (u)\approx u^{-u}.\,}
786:
774:
753:
714:
693:
657:
627:
562:
519:
515:
497:
459:
431:
427:
409:
347:
328:
297:
293:
275:
252:
205:
190:{\displaystyle u\rho '(u)+\rho (u-1)=0\,}
186:
137:
98:
1852:Arkiv för Matematik, Astronomi och Fysik
1012:
1841:
2036:"Integers without large prime factors"
2034:Hildebrand, A.; Tenenbaum, G. (1993).
1934:"On the number of positive integers â€
1900:"On the number of positive integers â€
726:{\displaystyle \rho (u)\approx u^{-u}}
1827:is controlled by the Dickman function
1269:{\displaystyle \rho _{n}(u)=\rho (u)}
678:{\displaystyle \Psi (x,y)=xu^{O(-u)}}
7:
2075:van de Lune, J.; Wattel, E. (1969).
616:and can be useful of its own right.
952:{\displaystyle e^{\xi }-1=u\xi .\,}
73:used to estimate the proportion of
2110:Bach, Eric; Peralta, René (1996).
1689:
1637:
1004:{\displaystyle \rho (x)\leq 1/x!.}
629:
499:
411:
349:
277:
25:
2173:10.1090/S0025-5718-1989-0969490-3
2096:10.1090/S0025-5718-1969-0247789-3
1359:{\displaystyle \rho (u)=1-\log u}
688:which is related to the estimate
447:{\displaystyle \Psi (x,x^{1/a})}
2012:10.1090/s0002-9904-1949-09337-0
1904:and free of prime factors >
748:A first approximation might be
93:The Dickmanâde Bruijn function
34:The Dickmanâde Bruijn function
1764:
1752:
1740:
1692:
1658:
1640:
1617:
1611:
1588:
1576:
1477:
1465:
1446:
1440:
1431:
1428:
1416:
1401:
1389:
1383:
1335:
1329:
1296:
1290:
1263:
1257:
1248:
1242:
1052:
1046:
978:
972:
886:
883:
877:
856:
816:
810:
764:
758:
704:
698:
670:
661:
644:
632:
576:
556:
547:
541:
529:
502:
473:
467:
441:
414:
364:
352:
325:
319:
307:
280:
216:
210:
177:
165:
156:
150:
109:
103:
1:
2140:10.1090/S0025-5718-96-00775-2
1820:{\displaystyle e^{-\gamma }}
1664:{\displaystyle \Psi (x,y,z)}
1594:{\displaystyle \sigma (u,v)}
128:delay differential equation
2331:
2160:Mathematics of Computation
2119:Mathematics of Computation
2082:Mathematics of Computation
1308:{\displaystyle \rho (u)=1}
370:{\displaystyle \Psi (x,y)}
247:Dickman proved that, when
228:{\displaystyle \rho (u)=1}
64:Dickmanâde Bruijn function
18:Dickmanâde Bruijn function
2258:10.1007/s11139-011-9304-3
1949:Indagationes Mathematicae
1932:de Bruijn, N. G. (1966).
1915:Indagationes Mathematicae
1898:de Bruijn, N. G. (1951).
1544:{\displaystyle \rho _{n}}
1218:{\displaystyle \rho _{n}}
479:{\displaystyle x\rho (a)}
83:Nicolaas Govert de Bruijn
2201:10.1112/plms/s3-33.3.565
1623:{\displaystyle \rho (u)}
1195:an integer, there is an
1191:For each interval with
1058:{\displaystyle \rho (u)}
962:A simple upper bound is
910:is the positive root of
200:with initial conditions
115:{\displaystyle \rho (u)}
1831:GolombâDickman constant
1797:, whose convergence to
738:GolombâDickman constant
2310:Analytic number theory
2189:Proc. London Math. Soc
1982:
1966:Ramaswami, V. (1949).
1821:
1775:
1665:
1624:
1595:
1545:
1507:
1360:
1309:
1270:
1219:
1059:
1029:
1005:
953:
893:
791:
727:
679:
609:
583:
480:
448:
393:) integers below
371:
333:
261:
229:
191:
116:
56:analytic number theory
51:
1983:
1878:Various (2012â2018).
1822:
1776:
1666:
1625:
1596:
1546:
1508:
1361:
1310:
1276:. For 0 â€
1271:
1220:
1060:
1030:
1006:
954:
894:
797:A better estimate is
792:
728:
680:
619:It can be shown that
607:
584:
481:
449:
372:
334:
262:
230:
192:
117:
33:
27:Mathematical function
1972:
1801:
1686:
1634:
1605:
1570:
1528:
1377:
1366:. For 2 â€
1323:
1315:. For 1 â€
1284:
1229:
1202:
1040:
1019:
966:
917:
904:exponential integral
804:
752:
692:
626:
496:
458:
408:
346:
274:
251:
204:
136:
97:
2131:1996MaCom..65.1701B
1865:1930ArMAF..22A..10D
126:that satisfies the
124:continuous function
2286:"Dickman function"
2283:Weisstein, Eric W.
2125:(216): 1701â1715.
1978:
1817:
1771:
1661:
1620:
1591:
1541:
1503:
1356:
1305:
1266:
1215:
1055:
1025:
1001:
949:
889:
787:
723:
675:
610:
579:
476:
454:was asymptotic to
444:
367:
329:
267:is fixed, we have
257:
235:for 0 â€
225:
187:
112:
52:
2315:Special functions
2235:Ramanujan Journal
2005:(12): 1122â1127.
1981:{\displaystyle x}
1791:Buchstab function
1498:
1197:analytic function
1184:
1183:
1028:{\displaystyle u}
845:
842:
377:is the number of
260:{\displaystyle a}
16:(Redirected from
2322:
2296:
2295:
2277:
2251:
2228:
2226:
2205:
2204:
2184:
2178:
2177:
2175:
2166:(187): 191â201.
2151:
2145:
2144:
2142:
2116:
2107:
2101:
2100:
2098:
2089:(106): 417â421.
2072:
2063:
2062:
2060:
2058:10.5802/jtnb.101
2040:
2031:
2025:
2024:
2014:
1996:
1987:
1985:
1984:
1979:
1963:
1957:
1956:
1946:
1929:
1923:
1922:
1912:
1895:
1889:
1887:
1875:
1869:
1868:
1846:
1826:
1824:
1823:
1818:
1816:
1815:
1780:
1778:
1777:
1772:
1739:
1738:
1734:
1718:
1717:
1713:
1670:
1668:
1667:
1662:
1629:
1627:
1626:
1621:
1600:
1598:
1597:
1592:
1556:trapezoidal rule
1550:
1548:
1547:
1542:
1540:
1539:
1512:
1510:
1509:
1504:
1499:
1494:
1493:
1484:
1461:
1460:
1370: †3,
1365:
1363:
1362:
1357:
1319: †2,
1314:
1312:
1311:
1306:
1280: †1,
1275:
1273:
1272:
1267:
1241:
1240:
1224:
1222:
1221:
1216:
1214:
1213:
1179:
1167:
1155:
1143:
1131:
1119:
1107:
1095:
1083:
1064:
1062:
1061:
1056:
1034:
1032:
1031:
1026:
1013:
1010:
1008:
1007:
1002:
991:
958:
956:
955:
950:
929:
928:
902:where Ei is the
898:
896:
895:
890:
846:
844:
843:
832:
823:
796:
794:
793:
788:
782:
781:
732:
730:
729:
724:
722:
721:
684:
682:
681:
676:
674:
673:
588:
586:
585:
580:
566:
528:
527:
523:
485:
483:
482:
477:
453:
451:
450:
445:
440:
439:
435:
376:
374:
373:
368:
338:
336:
335:
330:
306:
305:
301:
266:
264:
263:
258:
239: †1.
234:
232:
231:
226:
196:
194:
193:
188:
149:
121:
119:
118:
113:
71:special function
60:Dickman function
21:
2330:
2329:
2325:
2324:
2323:
2321:
2320:
2319:
2300:
2299:
2281:
2280:
2231:
2216:
2213:
2211:Further reading
2208:
2186:
2185:
2181:
2153:
2152:
2148:
2114:
2109:
2108:
2104:
2074:
2073:
2066:
2038:
2033:
2032:
2028:
1994:
1970:
1969:
1965:
1964:
1960:
1944:
1931:
1930:
1926:
1910:
1897:
1896:
1892:
1877:
1876:
1872:
1848:
1847:
1843:
1839:
1804:
1799:
1798:
1787:
1722:
1701:
1684:
1683:
1632:
1631:
1603:
1602:
1568:
1567:
1564:
1531:
1526:
1525:
1519:
1485:
1452:
1375:
1374:
1321:
1320:
1282:
1281:
1232:
1227:
1226:
1205:
1200:
1199:
1189:
1177:
1165:
1153:
1141:
1129:
1117:
1105:
1093:
1081:
1038:
1037:
1017:
1016:
964:
963:
920:
915:
914:
827:
802:
801:
770:
750:
749:
746:
710:
690:
689:
653:
624:
623:
602:
511:
494:
493:
456:
455:
423:
406:
405:
344:
343:
289:
272:
271:
249:
248:
245:
202:
201:
142:
134:
133:
95:
94:
91:
28:
23:
22:
15:
12:
11:
5:
2328:
2326:
2318:
2317:
2312:
2302:
2301:
2298:
2297:
2278:
2242:(1â3): 25â30.
2229:
2212:
2209:
2207:
2206:
2195:(3): 565â576.
2179:
2146:
2102:
2064:
2051:(2): 411â484.
2026:
1977:
1958:
1924:
1890:
1870:
1840:
1838:
1835:
1834:
1833:
1828:
1814:
1811:
1807:
1786:
1783:
1782:
1781:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1737:
1733:
1729:
1725:
1721:
1716:
1712:
1708:
1704:
1700:
1697:
1694:
1691:
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1619:
1616:
1613:
1610:
1590:
1587:
1584:
1581:
1578:
1575:
1563:
1560:
1538:
1534:
1517:
1514:
1513:
1502:
1497:
1492:
1488:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1459:
1455:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1355:
1352:
1349:
1346:
1343:
1340:
1337:
1334:
1331:
1328:
1304:
1301:
1298:
1295:
1292:
1289:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1239:
1235:
1212:
1208:
1188:
1185:
1182:
1181:
1174:
1170:
1169:
1162:
1158:
1157:
1150:
1146:
1145:
1138:
1134:
1133:
1126:
1122:
1121:
1114:
1110:
1109:
1102:
1098:
1097:
1090:
1086:
1085:
1078:
1074:
1073:
1070:
1066:
1065:
1054:
1051:
1048:
1045:
1035:
1024:
1000:
997:
994:
990:
986:
983:
980:
977:
974:
971:
960:
959:
947:
944:
941:
938:
935:
932:
927:
923:
900:
899:
888:
885:
882:
879:
876:
873:
870:
867:
864:
861:
858:
855:
852:
849:
841:
838:
835:
830:
826:
821:
818:
815:
812:
809:
785:
780:
777:
773:
769:
766:
763:
760:
757:
745:
742:
720:
717:
713:
709:
706:
703:
700:
697:
686:
685:
672:
669:
666:
663:
660:
656:
652:
649:
646:
643:
640:
637:
634:
631:
601:
598:
594:big O notation
590:
589:
578:
575:
572:
569:
565:
561:
558:
555:
552:
549:
546:
543:
540:
537:
534:
531:
526:
522:
518:
514:
510:
507:
504:
501:
475:
472:
469:
466:
463:
443:
438:
434:
430:
426:
422:
419:
416:
413:
366:
363:
360:
357:
354:
351:
340:
339:
327:
324:
321:
318:
315:
312:
309:
304:
300:
296:
292:
288:
285:
282:
279:
256:
244:
241:
224:
221:
218:
215:
212:
209:
198:
197:
185:
182:
179:
176:
173:
170:
167:
164:
161:
158:
155:
152:
148:
145:
141:
111:
108:
105:
102:
90:
87:
75:smooth numbers
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2327:
2316:
2313:
2311:
2308:
2307:
2305:
2293:
2292:
2287:
2284:
2279:
2275:
2271:
2267:
2263:
2259:
2255:
2250:
2245:
2241:
2237:
2236:
2230:
2225:
2220:
2215:
2214:
2210:
2202:
2198:
2194:
2190:
2183:
2180:
2174:
2169:
2165:
2161:
2157:
2150:
2147:
2141:
2136:
2132:
2128:
2124:
2120:
2113:
2106:
2103:
2097:
2092:
2088:
2084:
2083:
2078:
2071:
2069:
2065:
2059:
2054:
2050:
2046:
2045:
2037:
2030:
2027:
2022:
2018:
2013:
2008:
2004:
2000:
1993:
1991:
1975:
1962:
1959:
1954:
1950:
1943:
1941:
1937:
1928:
1925:
1920:
1916:
1909:
1907:
1903:
1894:
1891:
1885:
1881:
1874:
1871:
1866:
1862:
1858:
1854:
1853:
1845:
1842:
1836:
1832:
1829:
1812:
1809:
1805:
1796:
1795:rough numbers
1792:
1789:
1788:
1784:
1767:
1761:
1758:
1755:
1749:
1746:
1743:
1735:
1731:
1727:
1723:
1719:
1714:
1710:
1706:
1702:
1698:
1695:
1682:
1681:
1680:
1678:
1674:
1655:
1652:
1649:
1646:
1643:
1614:
1608:
1585:
1582:
1579:
1573:
1561:
1559:
1557:
1552:
1536:
1532:
1523:
1500:
1495:
1490:
1486:
1480:
1474:
1471:
1468:
1462:
1457:
1453:
1449:
1443:
1437:
1434:
1425:
1422:
1419:
1413:
1410:
1407:
1404:
1398:
1395:
1392:
1386:
1380:
1373:
1372:
1371:
1369:
1353:
1350:
1347:
1344:
1341:
1338:
1332:
1326:
1318:
1302:
1299:
1293:
1287:
1279:
1260:
1254:
1251:
1245:
1237:
1233:
1210:
1206:
1198:
1194:
1186:
1175:
1172:
1171:
1163:
1160:
1159:
1151:
1148:
1147:
1139:
1136:
1135:
1127:
1124:
1123:
1115:
1112:
1111:
1103:
1100:
1099:
1091:
1088:
1087:
1079:
1076:
1075:
1071:
1068:
1067:
1049:
1043:
1036:
1022:
1015:
1014:
1011:
998:
995:
992:
988:
984:
981:
975:
969:
945:
942:
939:
936:
933:
930:
925:
921:
913:
912:
911:
909:
905:
880:
874:
871:
868:
865:
862:
859:
853:
850:
847:
839:
836:
833:
828:
824:
819:
813:
807:
800:
799:
798:
783:
778:
775:
771:
767:
761:
755:
743:
741:
739:
734:
718:
715:
711:
707:
701:
695:
667:
664:
658:
654:
650:
647:
641:
638:
635:
622:
621:
620:
617:
615:
614:Pâ1 factoring
606:
599:
597:
595:
573:
570:
567:
563:
559:
553:
550:
544:
538:
535:
532:
524:
520:
516:
512:
508:
505:
492:
491:
490:
489:
470:
464:
461:
436:
432:
428:
424:
420:
417:
403:
398:
396:
392:
388:
384:
380:
361:
358:
355:
322:
316:
313:
310:
302:
298:
294:
290:
286:
283:
270:
269:
268:
254:
242:
240:
238:
222:
219:
213:
207:
183:
180:
174:
171:
168:
162:
159:
153:
146:
143:
139:
132:
131:
130:
129:
125:
106:
100:
88:
86:
84:
80:
76:
72:
68:
65:
61:
57:
49:
45:
41:
37:
32:
19:
2289:
2239:
2233:
2192:
2188:
2182:
2163:
2159:
2149:
2122:
2118:
2105:
2086:
2080:
2048:
2042:
2029:
2002:
1998:
1989:
1961:
1952:
1948:
1939:
1935:
1927:
1918:
1914:
1905:
1901:
1893:
1884:MathOverflow
1883:
1873:
1859:(10): 1â14.
1856:
1850:
1844:
1676:
1672:
1565:
1553:
1515:
1367:
1316:
1277:
1192:
1190:
961:
907:
901:
747:
735:
687:
618:
611:
600:Applications
591:
401:
399:
394:
386:
378:
341:
246:
236:
199:
92:
79:Karl Dickman
66:
63:
59:
53:
43:
39:
35:
1522:dilogarithm
1187:Computation
488:error bound
486:, with the
48:quasilinear
2304:Categories
1955:: 239â247.
1837:References
1225:such that
744:Estimation
243:Properties
89:Definition
2291:MathWorld
2274:119564455
2249:1005.3494
2224:1004.0519
1813:γ
1810:−
1750:σ
1744:∼
1690:Ψ
1638:Ψ
1609:ρ
1574:σ
1562:Extension
1533:ρ
1487:π
1472:−
1463:
1438:
1423:−
1414:
1408:−
1399:−
1381:ρ
1351:
1345:−
1327:ρ
1288:ρ
1255:ρ
1234:ρ
1207:ρ
1176:2.7701718
1164:1.0162483
1152:3.2320693
1140:8.7456700
1128:1.9649696
1116:3.5472470
1104:4.9109256
1092:4.8608388
1080:3.0685282
1044:ρ
982:≤
970:ρ
943:ξ
931:−
926:ξ
881:ξ
875:
866:ξ
860:−
854:
848:⋅
837:π
829:ξ
820:∼
808:ρ
776:−
768:≈
756:ρ
716:−
708:≈
696:ρ
665:−
630:Ψ
571:
539:ρ
500:Ψ
465:ρ
412:Ψ
350:Ψ
317:ρ
311:∼
278:Ψ
208:ρ
172:−
163:ρ
144:ρ
101:ρ
1921:: 50â60.
1785:See also
1679:. Then
1524:. Other
147:′
2266:2994087
2127:Bibcode
2021:0031958
1861:Bibcode
1516:with Li
733:below.
391:friable
2272:
2264:
2019:
383:smooth
342:where
58:, the
2270:S2CID
2244:arXiv
2219:arXiv
2115:(PDF)
2039:(PDF)
1995:(PDF)
1945:(PDF)
1942:, II"
1911:(PDF)
122:is a
69:is a
1520:the
906:and
736:The
385:(or
2254:doi
2197:doi
2168:doi
2135:doi
2091:doi
2053:doi
2007:doi
1857:22A
1601:of
1435:log
1411:log
1348:log
1180:10
1173:10
1168:10
1156:10
1144:10
1132:10
1120:10
1108:10
1096:10
1084:10
851:exp
592:in
568:log
62:or
54:In
2306::
2288:.
2268:.
2262:MR
2260:.
2252:.
2240:29
2238:.
2193:33
2191:.
2164:53
2162:.
2158:.
2133:.
2123:65
2121:.
2117:.
2087:23
2085:.
2079:.
2067:^
2047:.
2041:.
2017:MR
2015:.
2003:55
2001:.
1997:.
1953:28
1951:.
1947:.
1919:13
1917:.
1913:.
1882:.
1855:.
1496:12
1454:Li
1161:9
1149:8
1137:7
1125:6
1113:5
1101:4
1089:3
1077:2
1072:1
1069:1
872:Ei
596:.
404:,
397:.
85:.
2294:.
2276:.
2256::
2246::
2227:.
2221::
2203:.
2199::
2176:.
2170::
2143:.
2137::
2129::
2099:.
2093::
2061:.
2055::
2049:5
2023:.
2009::
1992:"
1990:x
1976:x
1940:y
1936:x
1908:"
1906:y
1902:x
1886:.
1867:.
1863::
1806:e
1768:.
1765:)
1762:a
1759:,
1756:b
1753:(
1747:x
1741:)
1736:b
1732:/
1728:1
1724:x
1720:,
1715:a
1711:/
1707:1
1703:x
1699:,
1696:x
1693:(
1677:z
1673:y
1659:)
1656:z
1653:,
1650:y
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1644:x
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1618:)
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1612:(
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1441:(
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1211:n
1193:n
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1166:Ă
1154:Ă
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1130:Ă
1118:Ă
1106:Ă
1094:Ă
1082:Ă
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1047:(
1023:u
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996:!
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940:u
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320:(
314:x
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303:a
299:/
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284:x
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