682:
744:
297:
619:
869:
806:
128:
222:
601:, the Wilf sequence is cited but with the initial terms switched. The resulting sequence appears primefree for the first hundred terms or so, but term 138 is the 45-digit prime
562:
467:
431:
635:
154:
582:
507:
487:
395:
359:
339:
903:
689:
306:
697:
250:
604:
597:
1170:
822:
759:
588:
1160:
76:
56:. To put it algebraically, a sequence of this type is defined by an appropriate choice of two composite numbers
71:
317:
that every term of this sequence is composite relies on the periodicity of
Fibonacci-like number sequences
162:
1165:
361:
repeat in a periodic pattern, and different primes in the set have overlapping patterns that result in a
510:
1085:
37:
519:
1021:
981:
949:
318:
314:
677:{\displaystyle a_{1}=331635635998274737472200656430763,a_{2}=1510028911088401971189590305498785}
926:
1133:
436:
400:
45:
584:. In this case, all the numbers in the sequence will be composite, but for a trivial reason.
1049:
1013:
973:
941:
133:
49:
41:
25:
1097:
1070:
1061:
993:
377:
is necessary for the question to be non-trivial. If the initial terms share a prime factor
1093:
1057:
989:
1089:
1034:
1136:
567:
514:
492:
472:
380:
344:
324:
1154:
922:
592:
229:
1111:
1017:
1001:
961:
751:
362:
241:
29:
893:
17:
1123:
564:
and more generally all subsequent values in the sequence will be multiples of
1141:
1118:
156:
there are no primes in the sequence of numbers calculated from the formula
33:
1025:
985:
953:
374:
53:
977:
945:
816:
The sequence of this type with the smallest known initial terms has
341:, the positions in the sequence where the numbers are divisible by
1053:
373:
The requirement that the initial terms of a primefree sequence be
739:{\displaystyle a_{1}=62638280004239857,a_{2}=49463435743205655}
292:{\displaystyle a_{1}=20615674205555510,a_{2}=3794765361567513}
614:{\displaystyle 439351292910452432574786963588089477522344721}
897:
872:
809:
747:
685:
301:
228:
The first primefree sequence of this type was published by
964:(1990). "A Fibonacci-like sequence of composite numbers".
321:
the members of a finite set of primes. For each prime
825:
762:
700:
638:
607:
587:
The order of the initial terms is also important. In
570:
522:
495:
475:
439:
403:
383:
347:
327:
253:
165:
136:
79:
1071:"A new Fibonacci-like sequence of composite numbers"
864:{\displaystyle a_{1}=106276436867,a_{2}=35256392432}
801:{\displaystyle a_{1}=407389224418,a_{2}=76343678551}
863:
800:
738:
676:
613:
576:
556:
501:
481:
461:
425:
389:
353:
333:
291:
216:
148:
122:
1035:"A Fibonacci-like sequence of composite numbers"
927:"A Fibonacci-like sequence of composite numbers"
1112:Problem 31. Fibonacci- all composites sequence
629:Several other primefree sequences are known:
609:439351292910452432574786963588089477522344721
8:
1114:. The prime puzzles and problems connection.
123:{\displaystyle \mathrm {gcd} (a_{1},a_{2})}
48:causing all members of the sequence to be
904:On-Line Encyclopedia of Integer Sequences
849:
830:
824:
786:
767:
761:
724:
705:
699:
662:
643:
637:
606:
569:
527:
521:
494:
474:
444:
438:
408:
402:
382:
346:
326:
277:
258:
252:
202:
183:
170:
164:
135:
111:
98:
80:
78:
32:. More specifically, it usually means a
885:
217:{\displaystyle a_{n}=a_{n-1}+a_{n-2}}
7:
1042:Electronic Journal of Combinatorics
672:1510028911088401971189590305498785
87:
84:
81:
14:
1004:(1990). "Letters to the Editor".
653:331635635998274737472200656430763
509:both greater than 1), due to the
130:is equal to 1, and such that for
1018:10.1080/0025570X.1990.11977539
598:The man who loved only numbers
548:
536:
240:A primefree sequence found by
117:
91:
52:that do not all have a common
1:
875:in the OEIS; Vsemirnov 2004).
1078:Journal of Integer Sequences
557:{\displaystyle a_{3}=(x+y)p}
1187:
894:Sloane, N. J. A.
812:in the OEIS; Nicol 1999).
28:that does not contain any
462:{\displaystyle a_{2}=yp}
426:{\displaystyle a_{1}=xp}
365:for the whole sequence.
235:
1033:Nicol, John W. (1999).
898:"Sequence A108156"
72:greatest common divisor
1069:Vsemirnov, M. (2004).
865:
802:
740:
678:
615:
578:
558:
503:
483:
463:
427:
391:
355:
335:
293:
218:
150:
149:{\displaystyle n>2}
124:
866:
803:
741:
679:
616:
579:
559:
511:distributive property
504:
484:
464:
428:
392:
356:
336:
294:
219:
151:
125:
44:, but with different
1171:Recurrence relations
1137:"Primefree Sequence"
1119:"Primefree sequence"
1006:Mathematics Magazine
966:Mathematics Magazine
934:Mathematics Magazine
823:
760:
698:
636:
605:
568:
520:
493:
473:
437:
401:
381:
345:
325:
251:
163:
134:
77:
36:defined by the same
26:sequence of integers
1090:2004JIntS...7...37V
244:has initial terms
38:recurrence relation
1134:Weisstein, Eric W.
907:. OEIS Foundation.
861:
798:
736:
674:
611:
574:
554:
499:
479:
459:
423:
387:
351:
331:
289:
214:
146:
120:
46:initial conditions
22:primefree sequence
1161:Integer sequences
923:Graham, Ronald L.
734:49463435743205655
715:62638280004239857
577:{\displaystyle p}
502:{\displaystyle y}
482:{\displaystyle x}
390:{\displaystyle p}
354:{\displaystyle p}
334:{\displaystyle p}
268:20615674205555510
50:composite numbers
42:Fibonacci numbers
1178:
1147:
1146:
1128:
1101:
1075:
1065:
1039:
1029:
1002:Wilf, Herbert S.
997:
962:Knuth, Donald E.
957:
931:
909:
908:
890:
870:
868:
867:
862:
854:
853:
835:
834:
807:
805:
804:
799:
791:
790:
772:
771:
745:
743:
742:
737:
729:
728:
710:
709:
683:
681:
680:
675:
667:
666:
648:
647:
620:
618:
617:
612:
591:'s biography of
583:
581:
580:
575:
563:
561:
560:
555:
532:
531:
508:
506:
505:
500:
488:
486:
485:
480:
468:
466:
465:
460:
449:
448:
432:
430:
429:
424:
413:
412:
396:
394:
393:
388:
360:
358:
357:
352:
340:
338:
337:
332:
304:
298:
296:
295:
290:
287:3794765361567513
282:
281:
263:
262:
223:
221:
220:
215:
213:
212:
194:
193:
175:
174:
155:
153:
152:
147:
129:
127:
126:
121:
116:
115:
103:
102:
90:
70:, such that the
1186:
1185:
1181:
1180:
1179:
1177:
1176:
1175:
1151:
1150:
1132:
1131:
1117:
1108:
1073:
1068:
1037:
1032:
1000:
978:10.2307/2691504
960:
946:10.2307/2689243
929:
921:
918:
913:
912:
892:
891:
887:
882:
845:
826:
821:
820:
782:
763:
758:
757:
720:
701:
696:
695:
692:; Graham 1964),
658:
639:
634:
633:
627:
625:Other sequences
603:
602:
566:
565:
523:
518:
517:
491:
490:
471:
470:
440:
435:
434:
404:
399:
398:
379:
378:
371:
343:
342:
323:
322:
300:
273:
254:
249:
248:
238:
236:Wilf's sequence
198:
179:
166:
161:
160:
132:
131:
107:
94:
75:
74:
69:
62:
12:
11:
5:
1184:
1182:
1174:
1173:
1168:
1163:
1153:
1152:
1149:
1148:
1129:
1115:
1107:
1106:External links
1104:
1103:
1102:
1066:
1030:
998:
958:
940:(5): 322–324.
917:
914:
911:
910:
884:
883:
881:
878:
877:
876:
860:
857:
852:
848:
844:
841:
838:
833:
829:
814:
813:
797:
794:
789:
785:
781:
778:
775:
770:
766:
755:
735:
732:
727:
723:
719:
716:
713:
708:
704:
693:
673:
670:
665:
661:
657:
654:
651:
646:
642:
626:
623:
610:
573:
553:
550:
547:
544:
541:
538:
535:
530:
526:
515:multiplication
498:
478:
458:
455:
452:
447:
443:
422:
419:
416:
411:
407:
386:
370:
367:
350:
330:
311:
310:
288:
285:
280:
276:
272:
269:
266:
261:
257:
237:
234:
226:
225:
211:
208:
205:
201:
197:
192:
189:
186:
182:
178:
173:
169:
145:
142:
139:
119:
114:
110:
106:
101:
97:
93:
89:
86:
83:
67:
60:
13:
10:
9:
6:
4:
3:
2:
1183:
1172:
1169:
1167:
1166:Number theory
1164:
1162:
1159:
1158:
1156:
1144:
1143:
1138:
1135:
1130:
1126:
1125:
1120:
1116:
1113:
1110:
1109:
1105:
1099:
1095:
1091:
1087:
1084:(3): 04.3.7.
1083:
1079:
1072:
1067:
1063:
1059:
1055:
1054:10.37236/1476
1051:
1047:
1043:
1036:
1031:
1027:
1023:
1019:
1015:
1011:
1007:
1003:
999:
995:
991:
987:
983:
979:
975:
971:
967:
963:
959:
955:
951:
947:
943:
939:
935:
928:
924:
920:
919:
915:
906:
905:
899:
895:
889:
886:
879:
874:
858:
855:
850:
846:
842:
839:
836:
831:
827:
819:
818:
817:
811:
795:
792:
787:
783:
779:
776:
773:
768:
764:
756:
753:
750:in the OEIS;
749:
733:
730:
725:
721:
717:
714:
711:
706:
702:
694:
691:
687:
671:
668:
663:
659:
655:
652:
649:
644:
640:
632:
631:
630:
624:
622:
608:
600:
599:
594:
590:
585:
571:
551:
545:
542:
539:
533:
528:
524:
516:
512:
496:
476:
456:
453:
450:
445:
441:
420:
417:
414:
409:
405:
384:
376:
369:Nontriviality
368:
366:
364:
348:
328:
320:
316:
308:
303:
286:
283:
278:
274:
270:
267:
264:
259:
255:
247:
246:
245:
243:
233:
231:
230:Ronald Graham
209:
206:
203:
199:
195:
190:
187:
184:
180:
176:
171:
167:
159:
158:
157:
143:
140:
137:
112:
108:
104:
99:
95:
73:
66:
59:
55:
51:
47:
43:
39:
35:
31:
30:prime numbers
27:
23:
19:
1140:
1122:
1081:
1077:
1045:
1041:
1009:
1005:
972:(1): 21–25.
969:
965:
937:
933:
901:
888:
840:106276436867
815:
777:407389224418
628:
596:
589:Paul Hoffman
586:
372:
363:covering set
312:
242:Herbert Wilf
239:
227:
64:
57:
21:
15:
859:35256392432
796:76343678551
397:(e.g., set
18:mathematics
1155:Categories
1124:PlanetMath
916:References
871:(sequence
808:(sequence
754:1990), and
746:(sequence
684:(sequence
593:Paul Erdős
299:(sequence
1142:MathWorld
1048:(1): 44.
469:for some
232:in 1964.
207:−
188:−
925:(1964).
34:sequence
1098:2110778
1086:Bibcode
1062:1728014
1026:2690956
1012:: 284.
994:1042933
986:2691504
954:2689243
896:(ed.).
873:A221286
810:A082411
748:A083105
688:in the
686:A083104
375:coprime
305:in the
302:A083216
54:divisor
40:as the
1096:
1060:
1024:
992:
984:
952:
319:modulo
1074:(PDF)
1038:(PDF)
1022:JSTOR
982:JSTOR
950:JSTOR
930:(PDF)
880:Notes
752:Knuth
315:proof
24:is a
902:The
690:OEIS
489:and
433:and
313:The
307:OEIS
141:>
63:and
20:, a
1050:doi
1014:doi
974:doi
942:doi
513:of
16:In
1157::
1139:.
1121:.
1094:MR
1092:.
1080:.
1076:.
1058:MR
1056:.
1044:.
1040:.
1020:.
1010:63
1008:.
990:MR
988:.
980:.
970:63
968:.
948:.
938:37
936:.
932:.
900:.
621:.
595:,
1145:.
1127:.
1100:.
1088::
1082:7
1064:.
1052::
1046:6
1028:.
1016::
996:.
976::
956:.
944::
856:=
851:2
847:a
843:,
837:=
832:1
828:a
793:=
788:2
784:a
780:,
774:=
769:1
765:a
731:=
726:2
722:a
718:,
712:=
707:1
703:a
669:=
664:2
660:a
656:,
650:=
645:1
641:a
572:p
552:p
549:)
546:y
543:+
540:x
537:(
534:=
529:3
525:a
497:y
477:x
457:p
454:y
451:=
446:2
442:a
421:p
418:x
415:=
410:1
406:a
385:p
349:p
329:p
309:)
284:=
279:2
275:a
271:,
265:=
260:1
256:a
224:.
210:2
204:n
200:a
196:+
191:1
185:n
181:a
177:=
172:n
168:a
144:2
138:n
118:)
113:2
109:a
105:,
100:1
96:a
92:(
88:d
85:c
82:g
68:2
65:a
61:1
58:a
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.