2934:
687:
that the prime-partitionable numbers appeared to be the same as the Erdős–Woods numbers, and this was proven in the same year by
Christopher Hunt Gribble. The same equivalence was also shown by Hasler and Mathar in 2015, together with an equivalence between two definitions of the prime-partitionable
629:
implied that such an element exists for every interval of at most 16 integers; thus, no Erdős–Woods number can be less than 16. In his 1981 thesis, Alan R. Woods independently conjectured that whenever
1036:
852:
718:
684:
350:
334:
683:, and they used a computer search to find several odd prime-partitionable numbers, including 15395 and 397197. In 2014, M. F. Hasler observed on the
1029:
972:
Hasler, Maximilian F.; Mathar, Richard J. (27 October 2015). "Corrigendum to "Paths and circuits in finite groups", Discr. Math. 22 (1978) 263".
817:
887:
1836:
1022:
622:
1831:
1846:
1826:
2539:
2119:
1841:
738:
2625:
883:"On the amplitude of intervals of natural numbers whose every element has a common prime divisor with at least an extremity"
1941:
667:
Meanwhile, the prime-partitionable numbers had been defined by
Holsztyński and Strube in 1978, following which Erdős and
2291:
1610:
1403:
621:
considered intervals of integers containing an element coprime to both endpoints. They observed that earlier results of
2326:
2296:
1971:
1961:
2467:
1881:
1615:
1595:
676:
2157:
2321:
2963:
2416:
2039:
1796:
1605:
1587:
1481:
1471:
1461:
2544:
2089:
1710:
1496:
1491:
1486:
1476:
1453:
671:
proved in 1978 that they form an infinite sequence. Erdős and
Trotter applied these results to generate pairs of
1529:
1786:
2655:
2620:
2406:
2316:
2190:
2165:
2074:
2064:
1676:
1658:
1578:
932:
86:
2958:
2915:
2185:
2059:
1690:
1466:
1246:
1173:
2170:
2024:
1951:
1106:
2879:
2519:
794:
Proceedings of the
Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1971)
2812:
2706:
2670:
2411:
2134:
2114:
1931:
1600:
1388:
1360:
2534:
2398:
2393:
2361:
2124:
2099:
2094:
2069:
1999:
1995:
1926:
1816:
1648:
1444:
1413:
2933:
826:
2937:
2691:
2686:
2600:
2574:
2472:
2451:
2223:
2104:
2054:
1976:
1946:
1886:
1653:
1633:
1564:
1277:
973:
916:
668:
343:
903, 2545, 4533, 5067, 8759, 9071, 9269, 10353, 11035, 11625, 11865, 13629, 15395, ... (sequence
340:
Although all of these initial numbers are even, odd Erdős–Woods numbers also exist. They include
1821:
2831:
2776:
2630:
2605:
2579:
2356:
2034:
2029:
1956:
1936:
1921:
1643:
1625:
1544:
1534:
1519:
1297:
1282:
680:
657:
2867:
2660:
2246:
2218:
2208:
2200:
2084:
2049:
2044:
2011:
1705:
1668:
1559:
1554:
1549:
1539:
1511:
1398:
1350:
1345:
1302:
1241:
941:
896:
861:
747:
24:
953:
801:
759:
2843:
2732:
2665:
2591:
2514:
2488:
2306:
2019:
1876:
1811:
1781:
1771:
1766:
1432:
1340:
1287:
1131:
1071:
949:
797:
755:
626:
2848:
2716:
2701:
2565:
2529:
2504:
2380:
2351:
2336:
2213:
2109:
2079:
1806:
1761:
1638:
1236:
1231:
1226:
1198:
1183:
1096:
1081:
1059:
1046:
782:
672:
618:
901:
882:
2952:
2771:
2755:
2696:
2650:
2346:
2331:
2241:
1966:
1524:
1393:
1355:
1312:
1193:
1178:
1168:
1126:
1116:
1091:
920:
778:
751:
661:
614:
62:
20:
641:
to both endpoints. It was only later that he found the first counterexample, , with
2807:
2796:
2711:
2549:
2524:
2441:
2341:
2311:
2286:
2270:
2175:
2142:
1891:
1865:
1776:
1715:
1292:
1188:
1121:
1101:
1076:
362:
361:
The Erdős–Woods numbers can be characterized in terms of certain partitions of the
324:
320:
316:
144:
736:
Holsztyński, W.; Strube, R. F. E. (1978). "Paths and circuits in finite groups".
2766:
2641:
2446:
1910:
1801:
1756:
1751:
1501:
1408:
1307:
1136:
1111:
1086:
1008:
708:
308:
304:
300:
296:
292:
288:
284:
280:
276:
272:
268:
264:
260:
256:
252:
248:
136:
648:. The existence of this counterexample shows that 16 is an Erdős–Woods number.
2903:
2884:
2180:
1791:
1004:
866:
847:
1014:
2509:
2436:
2428:
2233:
2147:
1265:
1000:
sequence A059757 (Initial terms of smallest Erdos-Woods intervals)
924:
786:
945:
2610:
39:
2615:
2274:
638:
369:
is an Erdős–Woods number if and only if the prime numbers less than
978:
34:
if it has the following property: there exists a positive integer
996:
381:
with the following property: for every pair of positive integers
139:
is an Erdős–Woods number because the 15 numbers between 2184 and
61:
of consecutive integers, each of the elements has a non-trivial
2901:
2865:
2829:
2793:
2753:
2378:
2267:
1993:
1908:
1863:
1740:
1430:
1377:
1329:
1263:
1215:
1153:
1057:
1018:
819:
Some problems in logic and number theory, and their connections
652:
proved that there are infinitely many Erdős–Woods numbers, and
925:"When the Cartesian product of directed cycles is Hamiltonian"
881:
Cégielski, Patrick; Heroult, François; Richard, Denis (2003).
312:
958:
See correction to the list of prime-partitionable numbers in
848:"On the existence of sequences of co-prime pairs of integers"
796:. Winnipeg, Manitoba: University of Manitoba. pp. 1–14.
764:
See correction to the list of prime-partitionable numbers in
69:
is an Erdős–Woods number if there exists a positive integer
999:
825:(PhD). University of Manchester. p. 88. Archived from
712:
345:
329:
653:
155:
These 15 numbers and their shared prime factor(s) are:
2725:
2679:
2639:
2590:
2564:
2497:
2481:
2460:
2427:
2392:
2232:
2199:
2156:
2133:
2010:
1698:
1689:
1667:
1624:
1586:
1577:
1510:
1452:
1443:
688:numbers from the two earlier works on the subject.
419:. For this reason, these numbers are also called
787:"Complete prime subsets of consecutive integers"
853:Journal of the Australian Mathematical Society
1030:
426:For instance, 16 is prime-partitionable with
8:
959:
765:
65:with one of the endpoints. In other words,
2898:
2862:
2826:
2790:
2750:
2424:
2389:
2375:
2264:
2007:
1990:
1905:
1860:
1737:
1695:
1583:
1449:
1440:
1427:
1374:
1331:Possessing a specific set of other numbers
1326:
1260:
1212:
1150:
1054:
1037:
1023:
1015:
977:
900:
865:
719:On-Line Encyclopedia of Integer Sequences
685:On-Line Encyclopedia of Integer Sequences
637:, the interval always includes a number
461:
158:
697:
654:Cégielski, Heroult & Richard (2003)
841:
839:
703:
701:
7:
731:
729:
649:
450:and corresponding prime divisors in
373:can be partitioned into two subsets
149:2184 = 2 · 3 · 7 · 13
245:The first Erdős–Woods numbers are
14:
2932:
2540:Perfect digit-to-digit invariant
440:}. The representations of 16 as
1:
1379:Expressible via specific sums
902:10.1016/S0304-3975(02)00444-9
888:Theoretical Computer Science
752:10.1016/0012-365X(78)90059-6
153:2200 = 2 · 5 · 11.
2468:Multiplicative digital root
677:Cartesian product of graphs
421:prime-partitionable numbers
415:is divisible by a prime in
407:is divisible by a prime in
73:such that for each integer
2980:
960:Hasler & Mathar (2015)
766:Hasler & Mathar (2015)
709:Sloane, N. J. A.
660:of Erdős–Woods numbers is
2928:
2911:
2897:
2875:
2861:
2839:
2825:
2803:
2789:
2762:
2749:
2545:Perfect digital invariant
2388:
2374:
2282:
2263:
2120:Superior highly composite
2006:
1989:
1917:
1904:
1872:
1859:
1747:
1736:
1439:
1426:
1384:
1373:
1336:
1325:
1273:
1259:
1222:
1211:
1164:
1149:
1067:
1053:
867:10.1017/S1446788700031220
2158:Euler's totient function
1942:Euler–Jacobi pseudoprime
1217:Other polynomial numbers
87:greatest common divisors
16:Type of positive integer
1972:Somer–Lucas pseudoprime
1962:Lucas–Carmichael number
1797:Lazy caterer's sequence
933:Journal of Graph Theory
917:Trotter, William T. Jr.
846:Dowe, David L. (1989).
816:Woods, Alan R. (1981).
713:"Sequence A059756"
1847:Wedderburn–Etherington
1247:Lucky numbers of Euler
946:10.1002/jgt.3190020206
85:, at least one of the
2135:Prime omega functions
1952:Frobenius pseudoprime
1742:Combinatorial numbers
1611:Centered dodecahedral
1404:Primary pseudoperfect
2594:-composition related
2394:Arithmetic functions
1996:Arithmetic functions
1932:Elliptic pseudoprime
1616:Centered icosahedral
1596:Centered tetrahedral
739:Discrete Mathematics
2520:Kaprekar's constant
2040:Colossally abundant
1927:Catalan pseudoprime
1827:Schröder–Hipparchus
1606:Centered octahedral
1482:Centered heptagonal
1472:Centered pentagonal
1462:Centered triangular
1062:and related numbers
1005:Erdős-Woods numbers
679:does not contain a
2938:Mathematics portal
2880:Aronson's sequence
2626:Smarandache–Wellin
2383:-dependent numbers
2090:Primitive abundant
1977:Strong pseudoprime
1967:Perrin pseudoprime
1947:Fermat pseudoprime
1887:Wolstenholme prime
1711:Squared triangular
1497:Centered decagonal
1492:Centered nonagonal
1487:Centered octagonal
1477:Centered hexagonal
722:. OEIS Foundation.
669:William T. Trotter
32:Erdős–Woods number
2964:Integer sequences
2946:
2945:
2924:
2923:
2893:
2892:
2857:
2856:
2821:
2820:
2785:
2784:
2745:
2744:
2741:
2740:
2560:
2559:
2370:
2369:
2259:
2258:
2255:
2254:
2201:Aliquot sequences
2012:Divisor functions
1985:
1984:
1957:Lucas pseudoprime
1937:Euler pseudoprime
1922:Carmichael number
1900:
1899:
1855:
1854:
1732:
1731:
1728:
1727:
1724:
1723:
1685:
1684:
1573:
1572:
1530:Square triangular
1422:
1421:
1369:
1368:
1321:
1320:
1255:
1254:
1207:
1206:
1145:
1144:
681:Hamiltonian cycle
635: > 1
613:In a 1971 paper,
605:
604:
242:
241:
38:such that in the
30:is said to be an
2971:
2936:
2899:
2868:Natural language
2863:
2827:
2795:Generated via a
2791:
2751:
2656:Digit-reassembly
2621:Self-descriptive
2425:
2390:
2376:
2327:Lucas–Carmichael
2317:Harmonic divisor
2265:
2191:Sparsely totient
2166:Highly cototient
2075:Multiply perfect
2065:Highly composite
2008:
1991:
1906:
1861:
1842:Telephone number
1738:
1696:
1677:Square pyramidal
1659:Stella octangula
1584:
1450:
1441:
1433:Figurate numbers
1428:
1375:
1327:
1261:
1213:
1151:
1055:
1039:
1032:
1025:
1016:
998:
984:
983:
981:
969:
963:
957:
929:
913:
907:
906:
904:
878:
872:
871:
869:
843:
834:
833:
831:
824:
813:
807:
805:
791:
783:Selfridge, J. L.
775:
769:
763:
733:
724:
723:
705:
656:showed that the
647:
636:
601:
593:
588:
580:
572:
567:
559:
549:
541:
533:
525:
517:
512:
502:
494:
486:
478:
470:
462:
457:
453:
449:
439:
432:
418:
414:
410:
406:
402:
388:
384:
380:
376:
372:
368:
357:Prime partitions
348:
332:
159:
154:
150:
142:
141:2200 = 2184 + 16
127:
124:is greater than
123:
103:
84:
80:
76:
72:
68:
60:
37:
29:
25:positive integer
2979:
2978:
2974:
2973:
2972:
2970:
2969:
2968:
2949:
2948:
2947:
2942:
2920:
2916:Strobogrammatic
2907:
2889:
2871:
2853:
2835:
2817:
2799:
2781:
2758:
2737:
2721:
2680:Divisor-related
2675:
2635:
2586:
2556:
2493:
2477:
2456:
2423:
2396:
2384:
2366:
2278:
2277:related numbers
2251:
2228:
2195:
2186:Perfect totient
2152:
2129:
2060:Highly abundant
2002:
1981:
1913:
1896:
1868:
1851:
1837:Stirling second
1743:
1720:
1681:
1663:
1620:
1569:
1506:
1467:Centered square
1435:
1418:
1380:
1365:
1332:
1317:
1269:
1268:defined numbers
1251:
1218:
1203:
1174:Double Mersenne
1160:
1141:
1063:
1049:
1047:natural numbers
1043:
1011:, June 30, 2024
993:
988:
987:
971:
970:
966:
927:
915:
914:
910:
880:
879:
875:
845:
844:
837:
829:
822:
815:
814:
810:
789:
777:
776:
772:
735:
734:
727:
707:
706:
699:
694:
673:directed cycles
646: = 16
642:
631:
627:George Szekeres
611:
606:
599:
597:
591:
589:
586:
584:
578:
576:
570:
568:
565:
563:
557:
555:
547:
545:
539:
537:
531:
529:
523:
521:
515:
513:
510:
508:
500:
498:
492:
490:
484:
482:
476:
474:
468:
466:
455:
451:
441:
434:
427:
416:
412:
408:
404:
390:
386:
382:
378:
374:
370:
366:
359:
354:
344:
338:
328:
243:
237:
232:
227:
222:
217:
210:
205:
200:
195:
190:
183:
178:
173:
168:
163:
152:
148:
140:
134:
125:
105:
89:
82:
78:
74:
70:
66:
42:
35:
27:
17:
12:
11:
5:
2977:
2975:
2967:
2966:
2961:
2951:
2950:
2944:
2943:
2941:
2940:
2929:
2926:
2925:
2922:
2921:
2919:
2918:
2912:
2909:
2908:
2902:
2895:
2894:
2891:
2890:
2888:
2887:
2882:
2876:
2873:
2872:
2866:
2859:
2858:
2855:
2854:
2852:
2851:
2849:Sorting number
2846:
2844:Pancake number
2840:
2837:
2836:
2830:
2823:
2822:
2819:
2818:
2816:
2815:
2810:
2804:
2801:
2800:
2794:
2787:
2786:
2783:
2782:
2780:
2779:
2774:
2769:
2763:
2760:
2759:
2756:Binary numbers
2754:
2747:
2746:
2743:
2742:
2739:
2738:
2736:
2735:
2729:
2727:
2723:
2722:
2720:
2719:
2714:
2709:
2704:
2699:
2694:
2689:
2683:
2681:
2677:
2676:
2674:
2673:
2668:
2663:
2658:
2653:
2647:
2645:
2637:
2636:
2634:
2633:
2628:
2623:
2618:
2613:
2608:
2603:
2597:
2595:
2588:
2587:
2585:
2584:
2583:
2582:
2571:
2569:
2566:P-adic numbers
2562:
2561:
2558:
2557:
2555:
2554:
2553:
2552:
2542:
2537:
2532:
2527:
2522:
2517:
2512:
2507:
2501:
2499:
2495:
2494:
2492:
2491:
2485:
2483:
2482:Coding-related
2479:
2478:
2476:
2475:
2470:
2464:
2462:
2458:
2457:
2455:
2454:
2449:
2444:
2439:
2433:
2431:
2422:
2421:
2420:
2419:
2417:Multiplicative
2414:
2403:
2401:
2386:
2385:
2381:Numeral system
2379:
2372:
2371:
2368:
2367:
2365:
2364:
2359:
2354:
2349:
2344:
2339:
2334:
2329:
2324:
2319:
2314:
2309:
2304:
2299:
2294:
2289:
2283:
2280:
2279:
2268:
2261:
2260:
2257:
2256:
2253:
2252:
2250:
2249:
2244:
2238:
2236:
2230:
2229:
2227:
2226:
2221:
2216:
2211:
2205:
2203:
2197:
2196:
2194:
2193:
2188:
2183:
2178:
2173:
2171:Highly totient
2168:
2162:
2160:
2154:
2153:
2151:
2150:
2145:
2139:
2137:
2131:
2130:
2128:
2127:
2122:
2117:
2112:
2107:
2102:
2097:
2092:
2087:
2082:
2077:
2072:
2067:
2062:
2057:
2052:
2047:
2042:
2037:
2032:
2027:
2025:Almost perfect
2022:
2016:
2014:
2004:
2003:
1994:
1987:
1986:
1983:
1982:
1980:
1979:
1974:
1969:
1964:
1959:
1954:
1949:
1944:
1939:
1934:
1929:
1924:
1918:
1915:
1914:
1909:
1902:
1901:
1898:
1897:
1895:
1894:
1889:
1884:
1879:
1873:
1870:
1869:
1864:
1857:
1856:
1853:
1852:
1850:
1849:
1844:
1839:
1834:
1832:Stirling first
1829:
1824:
1819:
1814:
1809:
1804:
1799:
1794:
1789:
1784:
1779:
1774:
1769:
1764:
1759:
1754:
1748:
1745:
1744:
1741:
1734:
1733:
1730:
1729:
1726:
1725:
1722:
1721:
1719:
1718:
1713:
1708:
1702:
1700:
1693:
1687:
1686:
1683:
1682:
1680:
1679:
1673:
1671:
1665:
1664:
1662:
1661:
1656:
1651:
1646:
1641:
1636:
1630:
1628:
1622:
1621:
1619:
1618:
1613:
1608:
1603:
1598:
1592:
1590:
1581:
1575:
1574:
1571:
1570:
1568:
1567:
1562:
1557:
1552:
1547:
1542:
1537:
1532:
1527:
1522:
1516:
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991:External links
989:
986:
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940:(2): 137–142.
908:
873:
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832:on 2019-06-08.
808:
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746:(3): 263–272.
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2115:Superabundant
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1389:Nonhypotenuse
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1128:
1127:Perfect power
1125:
1123:
1120:
1118:
1117:Seventh power
1115:
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397:
393:
364:
363:prime numbers
356:
352:
347:
341:
336:
331:
326:
322:
318:
314:
310:
306:
302:
298:
294:
290:
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282:
278:
274:
270:
266:
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258:
254:
250:
246:
235:
230:
225:
220:
215:
214:
208:
203:
198:
193:
188:
187:
181:
176:
171:
166:
161:
160:
156:
146:
143:each share a
138:
131:
129:
121:
117:
113:
109:
101:
97:
93:
88:
64:
63:common factor
58:
54:
50:
46:
41:
33:
26:
22:
21:number theory
2671:Transposable
2535:Narcissistic
2442:Digital root
2362:Super-Poulet
2322:Jordan–Pólya
2301:
2271:prime factor
2176:Noncototient
2143:Almost prime
2125:Superperfect
2100:Refactorable
2095:Quasiperfect
2070:Hyperperfect
1911:Pseudoprimes
1882:Wall–Sun–Sun
1817:Ordered Bell
1787:Fuss–Catalan
1699:non-centered
1649:Dodecahedral
1626:non-centered
1512:non-centered
1414:Wolstenholme
1159:× 2 ± 1
1156:
1155:Of the form
1122:Eighth power
1102:Fourth power
967:
937:
931:
911:
895:(1): 53–62.
892:
886:
876:
857:
851:
827:the original
818:
811:
793:
773:
743:
737:
716:
666:
643:
632:
623:S. S. Pillai
612:
514:2, 5 |
446:
442:
435:
428:
425:
420:
399:
395:
391:
360:
339:
327:… (sequence
244:
147:with one of
145:prime factor
135:
119:
115:
111:
107:
99:
95:
91:
56:
52:
48:
44:
31:
18:
2692:Extravagant
2687:Equidigital
2642:permutation
2601:Palindromic
2575:Automorphic
2473:Sum-product
2452:Sum-product
2407:Persistence
2302:Erdős–Woods
2224:Untouchable
2105:Semiperfect
2055:Hemiperfect
1716:Tesseractic
1654:Icosahedral
1634:Tetrahedral
1565:Dodecagonal
1266:Recursively
1137:Prime power
1112:Sixth power
1107:Fifth power
1087:Power of 10
1045:Classes of
1009:Numberphile
921:Erdős, Paul
650:Dowe (1989)
438:= {2, 5, 11
431:= {3, 7, 13
365:. A number
2959:Paul Erdős
2953:Categories
2904:Graphemics
2777:Pernicious
2631:Undulating
2606:Pandigital
2580:Trimorphic
2181:Nontotient
2030:Arithmetic
1644:Octahedral
1545:Heptagonal
1535:Pentagonal
1520:Triangular
1361:Sierpiński
1283:Jacobsthal
1082:Power of 3
1077:Power of 2
979:1510.07997
806:See p. 10.
692:References
615:Paul Erdős
577:13 |
499:11 |
2661:Parasitic
2510:Factorion
2437:Digit sum
2429:Digit sum
2247:Fortunate
2234:Primorial
2148:Semiprime
2085:Practical
2050:Descartes
2045:Deficient
2035:Betrothed
1877:Wieferich
1706:Pentatope
1669:pyramidal
1560:Decagonal
1555:Nonagonal
1550:Octagonal
1540:Hexagonal
1399:Practical
1346:Congruent
1278:Fibonacci
1242:Loeschian
860:: 84–89.
779:Erdős, P.
662:recursive
598:3 |
590:2 |
585:7 |
569:2 |
564:3 |
556:5 |
546:2 |
538:3 |
530:2 |
522:7 |
509:3 |
491:2 |
483:3 |
475:2 |
467:5 |
403:, either
2733:Friedman
2666:Primeval
2611:Repdigit
2568:-related
2515:Kaprekar
2489:Meertens
2412:Additive
2399:dynamics
2307:Friendly
2219:Sociable
2209:Amicable
2020:Abundant
2000:dynamics
1822:Schröder
1812:Narayana
1782:Eulerian
1772:Delannoy
1767:Dedekind
1588:centered
1454:centered
1341:Amenable
1298:Narayana
1288:Leonardo
1184:Mersenne
1132:Powerful
1072:Achilles
923:(1978).
785:(1971).
132:Examples
77:between
51:+ 1, …,
40:sequence
2906:related
2870:related
2834:related
2832:Sorting
2717:Vampire
2702:Harshad
2644:related
2616:Repunit
2530:Lychrel
2505:Dudeney
2357:Størmer
2352:Sphenic
2337:Regular
2275:divisor
2214:Perfect
2110:Sublime
2080:Perfect
1807:Motzkin
1762:Catalan
1303:Padovan
1237:Leyland
1232:Idoneal
1227:Hilbert
1199:Woodall
954:0493392
802:0337828
760:0522721
711:(ed.).
639:coprime
609:History
349:in the
346:A111042
333:in the
330:A059756
191:2, 3, 5
2772:Odious
2697:Frugal
2651:Cyclic
2640:Digit-
2347:Smooth
2332:Pronic
2292:Cyclic
2269:Other
2242:Euclid
1892:Wilson
1866:Primes
1525:Square
1394:Polite
1356:Riesel
1351:Knödel
1313:Perrin
1194:Thabit
1179:Fermat
1169:Cullen
1092:Square
1060:Powers
952:
800:
758:
675:whose
596:15 + 1
583:14 + 2
575:13 + 3
562:12 + 4
554:11 + 5
544:10 + 6
507:6 + 10
497:5 + 11
489:4 + 12
481:3 + 13
473:2 + 14
465:1 + 15
433:} and
2813:Prime
2808:Lucky
2797:sieve
2726:Other
2712:Smith
2592:Digit
2550:Happy
2525:Keith
2498:Other
2342:Rough
2312:Giuga
1777:Euler
1639:Cubic
1293:Lucas
1189:Proth
974:arXiv
928:(PDF)
830:(PDF)
823:(PDF)
790:(PDF)
536:9 + 7
528:8 + 8
520:7 + 9
458:are:
411:, or
389:with
2767:Evil
2447:Self
2397:and
2287:Blum
1998:and
1802:Lobb
1757:Cake
1752:Bell
1502:Star
1409:Ulam
1308:Pell
1097:Cube
997:OEIS
717:The
625:and
617:and
454:and
385:and
377:and
351:OEIS
335:OEIS
236:2199
233:2, 7
231:2198
226:2197
223:2, 3
221:2196
216:2195
209:2194
204:2193
199:2192
194:2191
189:2190
182:2189
177:2188
172:2187
167:2186
162:2185
151:and
106:gcd(
90:gcd(
81:and
23:, a
2885:Ban
2273:or
1792:Lah
942:doi
897:doi
893:303
862:doi
748:doi
658:set
325:116
321:112
317:106
313:100
104:or
19:In
2955::
1007:,
950:MR
948:.
936:.
930:.
919:;
891:.
885:.
858:47
856:.
850:.
838:^
798:MR
792:.
781:;
756:MR
754:.
744:22
742:.
728:^
715:.
700:^
664:.
445:+
423:.
398:=
394:+
353:).
337:).
323:,
319:,
315:,
311:,
309:96
307:,
305:94
303:,
301:92
299:,
297:88
295:,
293:86
291:,
289:78
287:,
285:76
283:,
281:70
279:,
277:66
275:,
273:64
271:,
269:56
267:,
265:46
263:,
261:36
259:,
257:34
255:,
253:22
251:,
249:16
228:13
184:11
137:16
128:.
118:+
114:,
110:+
98:+
94:,
55:+
47:,
1157:a
1038:e
1031:t
1024:v
982:.
976::
962:.
956:.
944::
938:2
905:.
899::
870:.
864::
804:.
768:.
762:.
750::
644:k
633:k
600:x
592:y
587:x
579:x
571:y
566:x
558:y
548:y
540:x
532:y
524:x
516:y
511:x
501:y
493:y
485:x
477:y
469:y
456:Y
452:X
447:y
443:x
436:Y
429:X
417:Y
413:y
409:X
405:x
400:k
396:y
392:x
387:y
383:x
379:Y
375:X
371:k
367:k
238:3
218:5
211:2
206:3
201:2
196:7
179:2
174:3
169:2
164:5
126:1
122:)
120:k
116:a
112:i
108:a
102:)
100:i
96:a
92:a
83:k
79:0
75:i
71:a
67:k
59:)
57:k
53:a
49:a
45:a
43:(
36:a
28:k
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