3449:
858:
594:
derived is 102040816326530612244897959183673469387755. Check the steps in Table One below. The algorithm begins building from right to left until it reaches step 15—then the infinite loop occurs. Lines 16 and 17 are pictured to show that nothing changes. There is a fix for this problem, and when applied, the algorithm will not only find all
593:
The step-by-step derivation algorithm depicted above is a great core technique but will not find all n-parasitic numbers. It will get stuck in an infinite loop when the derived number equals the derivation source. An example of this occurs when n = 5 and k = 5. The 42-digit n-parasitic number to be
797:
There is one more condition to be aware of when working with this algorithm, leading zeros must not be lost. When the shift number is created it may contain a leading zero which is positionally important and must be carried into and through the next step. Calculators and computer math methods will
270:
1308:
1, 105263157894736842, 1034482758620689655172413793, 102564, 102040816326530612244897959183673469387755, 1016949152542372881355932203389830508474576271186440677966, 1014492753623188405797, 1012658227848, 10112359550561797752808988764044943820224719, 10,
806: = 4. The Shift number created in step 4, 02564, has a leading zero which is fed into step 5 creating a leading zero product. The resulting Shift is fed into Step 6 which displays a product proving the 4-parasitic number ending in 4 is 102564.
1350:
1, 18, 28, 6, 42, 58, 22, 13, 44, 2, 108, 48, 21, 46, 148, 13, 78, 178, 6, 99, 18, 8, 228, 7, 41, 6, 268, 15, 272, 66, 34, 28, 138, 112, 116, 179, 5, 378, 388, 18, 204, 418, 6, 219, 32, 48, 66, 239, 81, 498, ... (sequence
173:
569:
517:
346:
438:
1551:
1309:
100917431192660550458715596330275229357798165137614678899082568807339449541284403669724770642201834862385321, 100840336134453781512605042016806722689075630252, ... (sequence
598:-parasitic numbers in base ten, it will find them in base 8 and base 16 as well. Look at line 15 in Table Two. The fix, when this condition is identified and the
3473:
1487:
1478:
1358:
1316:
884:
1544:
1071:-parasitic integers can be built by concatenation. For example, since 179487 is a 4-parasitic number, so are 179487179487, 179487179487179487 etc.
265:{\displaystyle x=0.179487179487179487\ldots =0.{\overline {179487}}{\mbox{ has }}4x=0.{\overline {717948}}={\frac {7.{\overline {179487}}}{10}}.}
528:
476:
281:
2351:
1537:
2346:
2361:
2341:
3054:
2634:
602:-parasitic number has not been found, is simply to not shift the product from the multiplication, but use it as is, and append
585:
105263157894736842 × 2 = 210526315789473684, which is the result of moving the last digit of 105263157894736842 to the front.
2356:
1376:
3140:
2456:
2806:
2125:
1918:
1087:-parasitic numbers are: (using inverted two and three for ten and eleven, respectively) (leading zeros are not allowed)
2841:
2811:
2486:
2476:
1418:
2982:
2396:
2130:
2110:
2672:
2836:
2931:
2554:
2311:
2120:
2102:
1996:
1986:
1976:
2816:
383:
3059:
2604:
2225:
2011:
2006:
2001:
1991:
1968:
2044:
48:
is itself a single-digit positive natural number. In other words, the decimal representation undergoes a right
2301:
3170:
3135:
2921:
2831:
2705:
2680:
2589:
2579:
2191:
2173:
2093:
1470:
1402:
3430:
2700:
2574:
2205:
1981:
1761:
1688:
1439:
2685:
2539:
2466:
1621:
161:
So 179487 is a 4-parasitic number with units digit 7. Others are 179487179487, 179487179487179487, etc.
3394:
3034:
3327:
3221:
3185:
2926:
2649:
2629:
2446:
2115:
1903:
1875:
1462:
445:
3049:
2913:
2908:
2876:
2639:
2614:
2609:
2584:
2514:
2510:
2441:
2331:
2163:
1959:
1928:
3448:
3452:
3206:
3201:
3115:
3089:
2987:
2966:
2738:
2619:
2569:
2491:
2461:
2401:
2168:
2148:
2079:
1792:
1519:
449:
91:
or greater) in the rightmost (units) place, and working up one digit at a time. For example, for
2336:
3346:
3291:
3145:
3120:
3094:
2871:
2549:
2544:
2471:
2451:
2436:
2158:
2140:
2059:
2049:
2034:
1812:
1797:
370:
165:
1397:
3382:
2761:
2733:
2723:
2715:
2599:
2564:
2559:
2526:
2220:
2183:
2074:
2069:
2064:
2054:
2026:
1913:
1865:
1860:
1817:
1756:
1503:
1515:
3358:
3247:
3180:
3106:
3029:
3003:
2821:
2534:
2391:
2326:
2296:
2286:
2281:
1947:
1855:
1802:
1646:
1586:
1511:
37:
606:(in this case 5) to the end. After 42 steps, the proper parasitic number will be found.
3363:
3231:
3216:
3080:
3044:
3019:
2895:
2866:
2851:
2728:
2624:
2594:
2321:
2276:
2153:
1751:
1746:
1741:
1713:
1698:
1611:
1596:
1574:
1561:
1444:
1423:
1332:
49:
29:
25:
3467:
3286:
3270:
3211:
3165:
2861:
2846:
2756:
2481:
2039:
1908:
1870:
1827:
1708:
1693:
1683:
1641:
1631:
1606:
1523:
1371:
874:
798:
remove leading zeros. Look at Table Three below displaying the derivation steps for
3322:
3311:
3226:
3064:
3039:
2956:
2856:
2826:
2801:
2785:
2690:
2657:
2406:
2380:
2291:
2230:
1807:
1703:
1636:
1616:
1591:
61:
3281:
3156:
2961:
2425:
2316:
2271:
2266:
2016:
1923:
1822:
1651:
1626:
1601:
3418:
3399:
2695:
2306:
1265:
1245:
1080:
1529:
1064:
then the numbers do not start with zero and hence fit the actual definition.
3024:
2951:
2943:
2748:
2662:
1780:
1048:
In general, if we relax the rules to allow a leading zero, then there are 9
791:
17. 5 × 59183673469387755 = 295918367346938775 − Shift = 959183673469387755
857:
3125:
786:
16. 5 × 9183673469387755 = 45918367346938775 − Shift = 59183673469387755
3130:
2789:
1507:
41:
781:
15. 5 × 183673469387755 = 918367346938775 − Shift = 9183673469387755
355:-parasitic number can be found as follows. Pick a one digit integer
1494:
Bernstein, Leon (1968), "Multiplicative twins and primitive roots",
697:
17. 5 × 183673469387755 = 918367346938775 − Shift = 183673469387755
692:
16. 5 × 183673469387755 = 918367346938775 − Shift = 183673469387755
687:
15. 5 × 183673469387755 = 918367346938775 − Shift = 183673469387755
776:
14. 5 × 83673469387755 = 418367346938775 − Shift = 183673469387755
682:
14. 5 × 83673469387755 = 418367346938775 − Shift = 183673469387755
564:{\displaystyle {\frac {2}{19}}=0.{\overline {105263157894736842}}.}
512:{\displaystyle {\frac {1}{19}}=0.{\overline {052631578947368421}}.}
341:{\displaystyle 4x={\frac {7+x}{10}}{\mbox{ so }}x={\frac {7}{39}}.}
1271:
1011235930336ᘔ53909ᘔ873Ɛ325819Ɛ9975055Ɛ54ᘔ3145ᘔ42694157078404491Ɛ
856:
67:
So even though 4 × 25641 = 102564, the number 25641 is
771:
13. 5 × 3673469387755 = 18367346938775 − Shift = 83673469387755
677:
13. 5 × 3673469387755 = 18367346938775 − Shift = 83673469387755
578:
of this period is 18, the same as the order of 10 modulo 19, so
3416:
3380:
3344:
3308:
3268:
2893:
2782:
2508:
2423:
2378:
2255:
1945:
1892:
1844:
1778:
1730:
1668:
1572:
1533:
766:
12. 5 × 673469387755 = 3367346938775 − Shift = 3673469387755
672:
12. 5 × 673469387755 = 3367346938775 − Shift = 3673469387755
1482:
1353:
1311:
989:
1016949152542372881355932203389830508474576271186440677966
879:
761:
11. 5 × 73469387755 = 367346938775 − Shift = 673469387755
667:
11. 5 × 73469387755 = 367346938775 − Shift = 673469387755
83:-parasitic number can be derived by starting with a digit
64:
to be used, and that is a commonly followed convention.
56:
4 × 128205 = 512820, so 128205 is 4-parasitic.
1300:
is obtained merely by shifting the leftmost digit 1 of
756:
10. 5 × 3469387755 = 17346938775 − Shift = 73469387755
662:
10. 5 × 3469387755 = 17346938775 − Shift = 73469387755
1234:
101419648634459Ɛ9384Ɛ26Ɛ533040547216ᘔ1155Ɛ3Ɛ12978ᘔ399
877:. They are: (leading zeros are not allowed) (sequence
313:
206:
531:
479:
386:
284:
176:
3240:
3194:
3154:
3105:
3079:
3012:
2996:
2975:
2942:
2907:
2747:
2714:
2671:
2648:
2525:
2213:
2204:
2182:
2139:
2101:
2092:
2025:
1967:
1958:
873:, after a puzzle concerning these numbers posed by
751:9. 5 × 469387755 = 2346938775 − Shift = 3469387755
657:9. 5 × 469387755 = 2346938775 − Shift = 3469387755
563:
511:
432:
340:
264:
746:8. 5 × 69387755 = 346938775 − Shift = 469387755
652:8. 5 × 69387755 = 346938775 − Shift = 469387755
470:− 1 = 19 and the repeating decimal for 1/19 is
1545:
1031:10112359550561797752808988764044943820224719
741:7. 5 × 9387755 = 46938775 − Shift = 69387755
647:7. 5 × 9387755 = 46938775 − Shift = 69387755
8:
433:{\displaystyle {\frac {k}{10n-1}}(10^{m}-1)}
3413:
3377:
3341:
3305:
3265:
2939:
2904:
2890:
2779:
2522:
2505:
2420:
2375:
2252:
2210:
2098:
1964:
1955:
1942:
1889:
1846:Possessing a specific set of other numbers
1841:
1775:
1727:
1665:
1569:
1552:
1538:
1530:
1203:101899Ɛ864406Ɛ33ᘔᘔ15423913745949305255Ɛ17
736:6. 5 × 387755 = 1938775 − Shift = 9387755
642:6. 5 × 387755 = 1938775 − Shift = 9387755
1488:On-Line Encyclopedia of Integer Sequences
548:
532:
530:
496:
480:
478:
415:
387:
385:
325:
312:
294:
283:
243:
237:
224:
205:
195:
175:
1292:beginning with 1 such that the quotient
1089:
889:
842:6. 4 × 102564 = 410256 − Shift = 102564
813:
707:
613:
1419:"Freeman Dyson's 4th-Grade Math Puzzle"
1388:
837:5. 4 × 02564 = 010256 − Shift = 102564
731:5. 5 × 87755 = 438775 − Shift = 387755
637:5. 5 × 87755 = 438775 − Shift = 387755
444:is the length of the period; i.e. the
1396:Dawidoff, Nicholas (March 25, 2009),
869:-parasitic numbers are also known as
7:
1189:1020408142854ᘔ997732650ᘔ18346916306
832:4. 4 × 2564 = 10256 − Shift = 02564
726:4. 5 × 7755 = 38775 − Shift = 87755
632:4. 5 × 7755 = 38775 − Shift = 87755
1288:In strict definition, least number
36:, results in movement of the last
14:
827:3. 4 × 564 = 2256 − Shift = 2564
721:3. 5 × 755 = 3775 − Shift = 7755
627:3. 5 × 755 = 3775 − Shift = 7755
522:So that for 2/19 is double that:
60:Most mathematicians do not allow
3474:Base-dependent integer sequences
3447:
3055:Perfect digit-to-digit invariant
1438:Tierney, John (April 13, 2009),
1417:Tierney, John (April 6, 2009),
1377:Linear-feedback shift register
1175:1025355ᘔ9433073ᘔ458409919Ɛ715
822:2. 4 × 64 = 256 − Shift = 564
716:2. 5 × 55 = 275 − Shift = 755
622:2. 5 × 55 = 275 − Shift = 755
427:
408:
1:
1894:Expressible via specific sums
1346:Number of digits of them are
1331:− 1), also the period of the
944:1034482758620689655172413793
369:, and take the period of the
52:by one place. For example:
1052:-parasitic numbers for each
553:
501:
248:
229:
200:
2983:Multiplicative digital root
817:1. 4 × 4 = 16 − Shift = 64
711:1. 5 × 5 = 25 − Shift = 55
617:1. 5 × 5 = 25 − Shift = 55
24:(in base 10) is a positive
3490:
87:(which should be equal to
3443:
3426:
3412:
3390:
3376:
3354:
3340:
3318:
3304:
3277:
3264:
3060:Perfect digital invariant
2903:
2889:
2797:
2778:
2635:Superior highly composite
2521:
2504:
2432:
2419:
2387:
2374:
2262:
2251:
1954:
1941:
1899:
1888:
1851:
1840:
1788:
1774:
1737:
1726:
1679:
1664:
1582:
1568:
1496:Mathematische Zeitschrift
2673:Euler's totient function
2457:Euler–Jacobi pseudoprime
1732:Other polynomial numbers
1440:"Prize for Dyson Puzzle"
1161:10309236ᘔ88206164719544
462:For another example, if
2487:Somer–Lucas pseudoprime
2477:Lucas–Carmichael number
2312:Lazy caterer's sequence
1471:Oxford University Press
1403:New York Times Magazine
1323:They are the period of
1003:1014492753623188405797
2362:Wedderburn–Etherington
1762:Lucky numbers of Euler
862:
589:Additional information
582:= 105263157894736842.
565:
513:
434:
342:
266:
42:decimal representation
2650:Prime omega functions
2467:Frobenius pseudoprime
2257:Combinatorial numbers
2126:Centered dodecahedral
1919:Primary pseudoperfect
1304:to the right end are
1083:system, the smallest
861:Freeman Dyson in 2005
860:
566:
514:
435:
343:
267:
3109:-composition related
2909:Arithmetic functions
2511:Arithmetic functions
2447:Elliptic pseudoprime
2131:Centered icosahedral
2111:Centered tetrahedral
1056:. Otherwise only if
529:
477:
446:multiplicative order
384:
282:
184:0.179487179487179487
174:
3035:Kaprekar's constant
2555:Colossally abundant
2442:Catalan pseudoprime
2342:Schröder–Hipparchus
2121:Centered octahedral
1997:Centered heptagonal
1987:Centered pentagonal
1977:Centered triangular
1577:and related numbers
1398:"The Civil Heretic"
930:105263157894736842
802: = 4 and
103:4 × 7 = 2
44:to its front. Here
3453:Mathematics portal
3395:Aronson's sequence
3141:Smarandache–Wellin
2898:-dependent numbers
2605:Primitive abundant
2492:Strong pseudoprime
2482:Perrin pseudoprime
2462:Fermat pseudoprime
2402:Wolstenholme prime
2226:Squared triangular
2012:Centered decagonal
2007:Centered nonagonal
2002:Centered octagonal
1992:Centered hexagonal
1508:10.1007/BF01135448
1467:Wonders of Numbers
1102:-parasitic number
902:-parasitic number
863:
853:-parasitic numbers
561:
551:105263157894736842
509:
499:052631578947368421
430:
380:−1). This will be
338:
317:
262:
210:
3461:
3460:
3439:
3438:
3408:
3407:
3372:
3371:
3336:
3335:
3300:
3299:
3260:
3259:
3256:
3255:
3075:
3074:
2885:
2884:
2774:
2773:
2770:
2769:
2716:Aliquot sequences
2527:Divisor functions
2500:
2499:
2472:Lucas pseudoprime
2452:Euler pseudoprime
2437:Carmichael number
2415:
2414:
2370:
2369:
2247:
2246:
2243:
2242:
2239:
2238:
2200:
2199:
2088:
2087:
2045:Square triangular
1937:
1936:
1884:
1883:
1836:
1835:
1770:
1769:
1722:
1721:
1660:
1659:
1284:Strict definition
1281:
1280:
1251:14Ɛ36429ᘔ7085792
1041:
1040:
846:
845:
795:
794:
701:
700:
556:
540:
504:
488:
406:
371:repeating decimal
333:
316:
310:
257:
251:
232:
209:
203:
166:repeating decimal
3481:
3451:
3414:
3383:Natural language
3378:
3342:
3310:Generated via a
3306:
3266:
3171:Digit-reassembly
3136:Self-descriptive
2940:
2905:
2891:
2842:Lucas–Carmichael
2832:Harmonic divisor
2780:
2706:Sparsely totient
2681:Highly cototient
2590:Multiply perfect
2580:Highly composite
2523:
2506:
2421:
2376:
2357:Telephone number
2253:
2211:
2192:Square pyramidal
2174:Stella octangula
2099:
1965:
1956:
1948:Figurate numbers
1943:
1890:
1842:
1776:
1728:
1666:
1570:
1554:
1547:
1540:
1531:
1526:
1485:
1450:
1448:
1435:
1429:
1427:
1414:
1408:
1406:
1393:
1356:
1314:
1090:
890:
882:
814:
708:
614:
581:
570:
568:
567:
562:
557:
549:
541:
533:
518:
516:
515:
510:
505:
497:
489:
481:
458:
439:
437:
436:
431:
420:
419:
407:
405:
388:
368:
347:
345:
344:
339:
334:
326:
318:
314:
311:
306:
295:
271:
269:
268:
263:
258:
253:
252:
244:
238:
233:
225:
211:
207:
204:
196:
164:Notice that the
22:parasitic number
3489:
3488:
3484:
3483:
3482:
3480:
3479:
3478:
3464:
3463:
3462:
3457:
3435:
3431:Strobogrammatic
3422:
3404:
3386:
3368:
3350:
3332:
3314:
3296:
3273:
3252:
3236:
3195:Divisor-related
3190:
3150:
3101:
3071:
3008:
2992:
2971:
2938:
2911:
2899:
2881:
2793:
2792:related numbers
2766:
2743:
2710:
2701:Perfect totient
2667:
2644:
2575:Highly abundant
2517:
2496:
2428:
2411:
2383:
2366:
2352:Stirling second
2258:
2235:
2196:
2178:
2135:
2084:
2021:
1982:Centered square
1950:
1933:
1895:
1880:
1847:
1832:
1784:
1783:defined numbers
1766:
1733:
1718:
1689:Double Mersenne
1675:
1656:
1578:
1564:
1562:natural numbers
1558:
1493:
1477:
1459:
1454:
1453:
1437:
1436:
1432:
1416:
1415:
1411:
1395:
1394:
1390:
1385:
1368:
1352:
1333:decadic integer
1310:
1286:
1077:
1046:
878:
855:
812:
706:
612:
591:
580:2 × (10 − 1)/19
579:
527:
526:
475:
474:
452:
411:
392:
382:
381:
360:
351:In general, an
296:
280:
279:
239:
208: has
172:
171:
77:
12:
11:
5:
3487:
3485:
3477:
3476:
3466:
3465:
3459:
3458:
3456:
3455:
3444:
3441:
3440:
3437:
3436:
3434:
3433:
3427:
3424:
3423:
3417:
3410:
3409:
3406:
3405:
3403:
3402:
3397:
3391:
3388:
3387:
3381:
3374:
3373:
3370:
3369:
3367:
3366:
3364:Sorting number
3361:
3359:Pancake number
3355:
3352:
3351:
3345:
3338:
3337:
3334:
3333:
3331:
3330:
3325:
3319:
3316:
3315:
3309:
3302:
3301:
3298:
3297:
3295:
3294:
3289:
3284:
3278:
3275:
3274:
3271:Binary numbers
3269:
3262:
3261:
3258:
3257:
3254:
3253:
3251:
3250:
3244:
3242:
3238:
3237:
3235:
3234:
3229:
3224:
3219:
3214:
3209:
3204:
3198:
3196:
3192:
3191:
3189:
3188:
3183:
3178:
3173:
3168:
3162:
3160:
3152:
3151:
3149:
3148:
3143:
3138:
3133:
3128:
3123:
3118:
3112:
3110:
3103:
3102:
3100:
3099:
3098:
3097:
3086:
3084:
3081:P-adic numbers
3077:
3076:
3073:
3072:
3070:
3069:
3068:
3067:
3057:
3052:
3047:
3042:
3037:
3032:
3027:
3022:
3016:
3014:
3010:
3009:
3007:
3006:
3000:
2998:
2997:Coding-related
2994:
2993:
2991:
2990:
2985:
2979:
2977:
2973:
2972:
2970:
2969:
2964:
2959:
2954:
2948:
2946:
2937:
2936:
2935:
2934:
2932:Multiplicative
2929:
2918:
2916:
2901:
2900:
2896:Numeral system
2894:
2887:
2886:
2883:
2882:
2880:
2879:
2874:
2869:
2864:
2859:
2854:
2849:
2844:
2839:
2834:
2829:
2824:
2819:
2814:
2809:
2804:
2798:
2795:
2794:
2783:
2776:
2775:
2772:
2771:
2768:
2767:
2765:
2764:
2759:
2753:
2751:
2745:
2744:
2742:
2741:
2736:
2731:
2726:
2720:
2718:
2712:
2711:
2709:
2708:
2703:
2698:
2693:
2688:
2686:Highly totient
2683:
2677:
2675:
2669:
2668:
2666:
2665:
2660:
2654:
2652:
2646:
2645:
2643:
2642:
2637:
2632:
2627:
2622:
2617:
2612:
2607:
2602:
2597:
2592:
2587:
2582:
2577:
2572:
2567:
2562:
2557:
2552:
2547:
2542:
2540:Almost perfect
2537:
2531:
2529:
2519:
2518:
2509:
2502:
2501:
2498:
2497:
2495:
2494:
2489:
2484:
2479:
2474:
2469:
2464:
2459:
2454:
2449:
2444:
2439:
2433:
2430:
2429:
2424:
2417:
2416:
2413:
2412:
2410:
2409:
2404:
2399:
2394:
2388:
2385:
2384:
2379:
2372:
2371:
2368:
2367:
2365:
2364:
2359:
2354:
2349:
2347:Stirling first
2344:
2339:
2334:
2329:
2324:
2319:
2314:
2309:
2304:
2299:
2294:
2289:
2284:
2279:
2274:
2269:
2263:
2260:
2259:
2256:
2249:
2248:
2245:
2244:
2241:
2240:
2237:
2236:
2234:
2233:
2228:
2223:
2217:
2215:
2208:
2202:
2201:
2198:
2197:
2195:
2194:
2188:
2186:
2180:
2179:
2177:
2176:
2171:
2166:
2161:
2156:
2151:
2145:
2143:
2137:
2136:
2134:
2133:
2128:
2123:
2118:
2113:
2107:
2105:
2096:
2090:
2089:
2086:
2085:
2083:
2082:
2077:
2072:
2067:
2062:
2057:
2052:
2047:
2042:
2037:
2031:
2029:
2023:
2022:
2020:
2019:
2014:
2009:
2004:
1999:
1994:
1989:
1984:
1979:
1973:
1971:
1962:
1952:
1951:
1946:
1939:
1938:
1935:
1934:
1932:
1931:
1926:
1921:
1916:
1911:
1906:
1900:
1897:
1896:
1893:
1886:
1885:
1882:
1881:
1879:
1878:
1873:
1868:
1863:
1858:
1852:
1849:
1848:
1845:
1838:
1837:
1834:
1833:
1831:
1830:
1825:
1820:
1815:
1810:
1805:
1800:
1795:
1789:
1786:
1785:
1779:
1772:
1771:
1768:
1767:
1765:
1764:
1759:
1754:
1749:
1744:
1738:
1735:
1734:
1731:
1724:
1723:
1720:
1719:
1717:
1716:
1711:
1706:
1701:
1696:
1691:
1686:
1680:
1677:
1676:
1669:
1662:
1661:
1658:
1657:
1655:
1654:
1649:
1644:
1639:
1634:
1629:
1624:
1619:
1614:
1609:
1604:
1599:
1594:
1589:
1583:
1580:
1579:
1573:
1566:
1565:
1559:
1557:
1556:
1549:
1542:
1534:
1528:
1527:
1491:
1474:
1469:, Chapter 28,
1463:C. A. Pickover
1458:
1455:
1452:
1451:
1445:New York Times
1430:
1424:New York Times
1409:
1387:
1386:
1384:
1381:
1380:
1379:
1374:
1367:
1364:
1363:
1362:
1321:
1320:
1285:
1282:
1279:
1278:
1275:
1272:
1269:
1262:
1261:
1255:
1252:
1249:
1242:
1241:
1238:
1235:
1232:
1228:
1227:
1221:
1218:
1215:
1211:
1210:
1207:
1204:
1201:
1197:
1196:
1193:
1190:
1187:
1183:
1182:
1179:
1176:
1173:
1169:
1168:
1165:
1162:
1159:
1155:
1154:
1148:
1145:
1142:
1138:
1137:
1134:
1131:
1128:
1124:
1123:
1120:
1117:
1114:
1110:
1109:
1106:
1103:
1096:
1076:
1073:
1045:
1042:
1039:
1038:
1035:
1032:
1029:
1025:
1024:
1021:
1018:
1017:1012658227848
1015:
1011:
1010:
1007:
1004:
1001:
997:
996:
993:
990:
987:
983:
982:
976:
973:
970:
966:
965:
962:
959:
956:
952:
951:
948:
945:
942:
938:
937:
934:
931:
928:
924:
923:
920:
917:
914:
910:
909:
906:
903:
896:
854:
847:
844:
843:
839:
838:
834:
833:
829:
828:
824:
823:
819:
818:
811:
808:
793:
792:
788:
787:
783:
782:
778:
777:
773:
772:
768:
767:
763:
762:
758:
757:
753:
752:
748:
747:
743:
742:
738:
737:
733:
732:
728:
727:
723:
722:
718:
717:
713:
712:
705:
702:
699:
698:
694:
693:
689:
688:
684:
683:
679:
678:
674:
673:
669:
668:
664:
663:
659:
658:
654:
653:
649:
648:
644:
643:
639:
638:
634:
633:
629:
628:
624:
623:
619:
618:
611:
608:
590:
587:
572:
571:
560:
555:
552:
547:
544:
539:
536:
520:
519:
508:
503:
500:
495:
492:
487:
484:
429:
426:
423:
418:
414:
410:
404:
401:
398:
395:
391:
349:
348:
337:
332:
329:
324:
321:
315: so
309:
305:
302:
299:
293:
290:
287:
273:
272:
261:
256:
250:
247:
242:
236:
231:
228:
223:
220:
217:
214:
202:
199:
194:
191:
188:
185:
182:
179:
159:
158:
149:4 ×
147:
139:4 ×
137:
129:4 ×
127:
119:4 ×
117:
109:4 ×
107:
76:
73:
58:
57:
50:circular shift
26:natural number
13:
10:
9:
6:
4:
3:
2:
3486:
3475:
3472:
3471:
3469:
3454:
3450:
3446:
3445:
3442:
3432:
3429:
3428:
3425:
3420:
3415:
3411:
3401:
3398:
3396:
3393:
3392:
3389:
3384:
3379:
3375:
3365:
3362:
3360:
3357:
3356:
3353:
3348:
3343:
3339:
3329:
3326:
3324:
3321:
3320:
3317:
3313:
3307:
3303:
3293:
3290:
3288:
3285:
3283:
3280:
3279:
3276:
3272:
3267:
3263:
3249:
3246:
3245:
3243:
3239:
3233:
3230:
3228:
3225:
3223:
3222:Polydivisible
3220:
3218:
3215:
3213:
3210:
3208:
3205:
3203:
3200:
3199:
3197:
3193:
3187:
3184:
3182:
3179:
3177:
3174:
3172:
3169:
3167:
3164:
3163:
3161:
3158:
3153:
3147:
3144:
3142:
3139:
3137:
3134:
3132:
3129:
3127:
3124:
3122:
3119:
3117:
3114:
3113:
3111:
3108:
3104:
3096:
3093:
3092:
3091:
3088:
3087:
3085:
3082:
3078:
3066:
3063:
3062:
3061:
3058:
3056:
3053:
3051:
3048:
3046:
3043:
3041:
3038:
3036:
3033:
3031:
3028:
3026:
3023:
3021:
3018:
3017:
3015:
3011:
3005:
3002:
3001:
2999:
2995:
2989:
2986:
2984:
2981:
2980:
2978:
2976:Digit product
2974:
2968:
2965:
2963:
2960:
2958:
2955:
2953:
2950:
2949:
2947:
2945:
2941:
2933:
2930:
2928:
2925:
2924:
2923:
2920:
2919:
2917:
2915:
2910:
2906:
2902:
2897:
2892:
2888:
2878:
2875:
2873:
2870:
2868:
2865:
2863:
2860:
2858:
2855:
2853:
2850:
2848:
2845:
2843:
2840:
2838:
2835:
2833:
2830:
2828:
2825:
2823:
2820:
2818:
2815:
2813:
2812:Erdős–Nicolas
2810:
2808:
2805:
2803:
2800:
2799:
2796:
2791:
2787:
2781:
2777:
2763:
2760:
2758:
2755:
2754:
2752:
2750:
2746:
2740:
2737:
2735:
2732:
2730:
2727:
2725:
2722:
2721:
2719:
2717:
2713:
2707:
2704:
2702:
2699:
2697:
2694:
2692:
2689:
2687:
2684:
2682:
2679:
2678:
2676:
2674:
2670:
2664:
2661:
2659:
2656:
2655:
2653:
2651:
2647:
2641:
2638:
2636:
2633:
2631:
2630:Superabundant
2628:
2626:
2623:
2621:
2618:
2616:
2613:
2611:
2608:
2606:
2603:
2601:
2598:
2596:
2593:
2591:
2588:
2586:
2583:
2581:
2578:
2576:
2573:
2571:
2568:
2566:
2563:
2561:
2558:
2556:
2553:
2551:
2548:
2546:
2543:
2541:
2538:
2536:
2533:
2532:
2530:
2528:
2524:
2520:
2516:
2512:
2507:
2503:
2493:
2490:
2488:
2485:
2483:
2480:
2478:
2475:
2473:
2470:
2468:
2465:
2463:
2460:
2458:
2455:
2453:
2450:
2448:
2445:
2443:
2440:
2438:
2435:
2434:
2431:
2427:
2422:
2418:
2408:
2405:
2403:
2400:
2398:
2395:
2393:
2390:
2389:
2386:
2382:
2377:
2373:
2363:
2360:
2358:
2355:
2353:
2350:
2348:
2345:
2343:
2340:
2338:
2335:
2333:
2330:
2328:
2325:
2323:
2320:
2318:
2315:
2313:
2310:
2308:
2305:
2303:
2300:
2298:
2295:
2293:
2290:
2288:
2285:
2283:
2280:
2278:
2275:
2273:
2270:
2268:
2265:
2264:
2261:
2254:
2250:
2232:
2229:
2227:
2224:
2222:
2219:
2218:
2216:
2212:
2209:
2207:
2206:4-dimensional
2203:
2193:
2190:
2189:
2187:
2185:
2181:
2175:
2172:
2170:
2167:
2165:
2162:
2160:
2157:
2155:
2152:
2150:
2147:
2146:
2144:
2142:
2138:
2132:
2129:
2127:
2124:
2122:
2119:
2117:
2116:Centered cube
2114:
2112:
2109:
2108:
2106:
2104:
2100:
2097:
2095:
2094:3-dimensional
2091:
2081:
2078:
2076:
2073:
2071:
2068:
2066:
2063:
2061:
2058:
2056:
2053:
2051:
2048:
2046:
2043:
2041:
2038:
2036:
2033:
2032:
2030:
2028:
2024:
2018:
2015:
2013:
2010:
2008:
2005:
2003:
2000:
1998:
1995:
1993:
1990:
1988:
1985:
1983:
1980:
1978:
1975:
1974:
1972:
1970:
1966:
1963:
1961:
1960:2-dimensional
1957:
1953:
1949:
1944:
1940:
1930:
1927:
1925:
1922:
1920:
1917:
1915:
1912:
1910:
1907:
1905:
1904:Nonhypotenuse
1902:
1901:
1898:
1891:
1887:
1877:
1874:
1872:
1869:
1867:
1864:
1862:
1859:
1857:
1854:
1853:
1850:
1843:
1839:
1829:
1826:
1824:
1821:
1819:
1816:
1814:
1811:
1809:
1806:
1804:
1801:
1799:
1796:
1794:
1791:
1790:
1787:
1782:
1777:
1773:
1763:
1760:
1758:
1755:
1753:
1750:
1748:
1745:
1743:
1740:
1739:
1736:
1729:
1725:
1715:
1712:
1710:
1707:
1705:
1702:
1700:
1697:
1695:
1692:
1690:
1687:
1685:
1682:
1681:
1678:
1673:
1667:
1663:
1653:
1650:
1648:
1645:
1643:
1642:Perfect power
1640:
1638:
1635:
1633:
1632:Seventh power
1630:
1628:
1625:
1623:
1620:
1618:
1615:
1613:
1610:
1608:
1605:
1603:
1600:
1598:
1595:
1593:
1590:
1588:
1585:
1584:
1581:
1576:
1571:
1567:
1563:
1555:
1550:
1548:
1543:
1541:
1536:
1535:
1532:
1525:
1521:
1517:
1513:
1509:
1505:
1501:
1497:
1492:
1489:
1484:
1480:
1475:
1472:
1468:
1464:
1461:
1460:
1456:
1447:
1446:
1441:
1434:
1431:
1426:
1425:
1420:
1413:
1410:
1405:
1404:
1399:
1392:
1389:
1382:
1378:
1375:
1373:
1372:Cyclic number
1370:
1369:
1365:
1360:
1355:
1349:
1348:
1347:
1344:
1342:
1338:
1334:
1330:
1326:
1318:
1313:
1307:
1306:
1305:
1303:
1299:
1295:
1291:
1283:
1276:
1273:
1270:
1267:
1264:
1263:
1259:
1256:
1253:
1250:
1247:
1244:
1243:
1239:
1236:
1233:
1230:
1229:
1225:
1222:
1219:
1216:
1213:
1212:
1208:
1205:
1202:
1199:
1198:
1194:
1191:
1188:
1185:
1184:
1180:
1177:
1174:
1171:
1170:
1166:
1163:
1160:
1157:
1156:
1152:
1149:
1146:
1143:
1140:
1139:
1135:
1132:
1129:
1126:
1125:
1121:
1118:
1115:
1112:
1111:
1107:
1104:
1101:
1097:
1095:
1092:
1091:
1088:
1086:
1082:
1074:
1072:
1070:
1065:
1063:
1059:
1055:
1051:
1043:
1036:
1033:
1030:
1027:
1026:
1022:
1019:
1016:
1013:
1012:
1008:
1005:
1002:
999:
998:
994:
991:
988:
985:
984:
980:
977:
974:
971:
968:
967:
963:
960:
957:
954:
953:
949:
946:
943:
940:
939:
935:
932:
929:
926:
925:
921:
918:
915:
912:
911:
907:
904:
901:
897:
895:
892:
891:
888:
886:
881:
876:
875:Freeman Dyson
872:
871:Dyson numbers
868:
865:The smallest
859:
852:
848:
841:
840:
836:
835:
831:
830:
826:
825:
821:
820:
816:
815:
809:
807:
805:
801:
790:
789:
785:
784:
780:
779:
775:
774:
770:
769:
765:
764:
760:
759:
755:
754:
750:
749:
745:
744:
740:
739:
735:
734:
730:
729:
725:
724:
720:
719:
715:
714:
710:
709:
703:
696:
695:
691:
690:
686:
685:
681:
680:
676:
675:
671:
670:
666:
665:
661:
660:
656:
655:
651:
650:
646:
645:
641:
640:
636:
635:
631:
630:
626:
625:
621:
620:
616:
615:
609:
607:
605:
601:
597:
588:
586:
583:
577:
558:
550:
545:
542:
537:
534:
525:
524:
523:
506:
498:
493:
490:
485:
482:
473:
472:
471:
469:
465:
460:
456:
451:
447:
443:
424:
421:
416:
412:
402:
399:
396:
393:
389:
379:
375:
372:
367:
363:
358:
354:
335:
330:
327:
322:
319:
307:
303:
300:
297:
291:
288:
285:
278:
277:
276:
259:
254:
245:
240:
234:
226:
221:
218:
215:
212:
197:
192:
189:
186:
183:
180:
177:
170:
169:
168:
167:
162:
156:
152:
148:
146:
142:
138:
136:
132:
128:
126:
122:
118:
116:
112:
108:
106:
102:
101:
100:
98:
94:
90:
86:
82:
74:
72:
71:4-parasitic.
70:
65:
63:
62:leading zeros
55:
54:
53:
51:
47:
43:
39:
35:
31:
27:
23:
19:
3186:Transposable
3175:
3050:Narcissistic
2957:Digital root
2877:Super-Poulet
2837:Jordan–Pólya
2786:prime factor
2691:Noncototient
2658:Almost prime
2640:Superperfect
2615:Refactorable
2610:Quasiperfect
2585:Hyperperfect
2426:Pseudoprimes
2397:Wall–Sun–Sun
2332:Ordered Bell
2302:Fuss–Catalan
2214:non-centered
2164:Dodecahedral
2141:non-centered
2027:non-centered
1929:Wolstenholme
1674:× 2 ± 1
1671:
1670:Of the form
1637:Eighth power
1617:Fourth power
1499:
1495:
1466:
1443:
1433:
1422:
1412:
1401:
1391:
1345:
1340:
1336:
1328:
1324:
1322:
1301:
1297:
1293:
1289:
1287:
1257:
1223:
1150:
1130:10631694842
1099:
1093:
1084:
1078:
1068:
1066:
1061:
1057:
1053:
1049:
1047:
1044:General note
978:
899:
893:
870:
866:
864:
850:
803:
799:
796:
603:
599:
595:
592:
584:
575:
573:
521:
467:
466:= 2, then 10
463:
461:
454:
441:
377:
373:
365:
361:
356:
352:
350:
274:
163:
160:
154:
150:
144:
140:
134:
130:
124:
120:
114:
110:
104:
96:
92:
88:
84:
80:
78:
68:
66:
59:
45:
33:
28:which, when
21:
17:
15:
3207:Extravagant
3202:Equidigital
3157:permutation
3116:Palindromic
3090:Automorphic
2988:Sum-product
2967:Sum-product
2922:Persistence
2817:Erdős–Woods
2739:Untouchable
2620:Semiperfect
2570:Hemiperfect
2231:Tesseractic
2169:Icosahedral
2149:Tetrahedral
2080:Dodecagonal
1781:Recursively
1652:Prime power
1627:Sixth power
1622:Fifth power
1602:Power of 10
1560:Classes of
1260:/9Ɛ = 2/15
1226:/7Ɛ = 2/17
1075:Other bases
810:Table Three
574:The length
3419:Graphemics
3292:Pernicious
3146:Undulating
3121:Pandigital
3095:Trimorphic
2696:Nontotient
2545:Arithmetic
2159:Octahedral
2060:Heptagonal
2050:Pentagonal
2035:Triangular
1876:Sierpiński
1798:Jacobsthal
1597:Power of 3
1592:Power of 2
1457:References
1153:/2Ɛ = 1/5
1108:Period of
1081:duodecimal
981:/49 = 1/7
908:Period of
359:such that
75:Derivation
30:multiplied
3176:Parasitic
3025:Factorion
2952:Digit sum
2944:Digit sum
2762:Fortunate
2749:Primorial
2663:Semiprime
2600:Practical
2565:Descartes
2560:Deficient
2550:Betrothed
2392:Wieferich
2221:Pentatope
2184:pyramidal
2075:Decagonal
2070:Nonagonal
2065:Octagonal
2055:Hexagonal
1914:Practical
1861:Congruent
1793:Fibonacci
1757:Loeschian
1524:121138247
1502:: 49–58,
1476:Sequence
1473:UK, 2000.
1098:Smallest
898:Smallest
849:Smallest
704:Table Two
610:Table One
554:¯
502:¯
422:−
400:−
249:¯
230:¯
201:¯
187:…
3468:Category
3248:Friedman
3181:Primeval
3126:Repdigit
3083:-related
3030:Kaprekar
3004:Meertens
2927:Additive
2914:dynamics
2822:Friendly
2734:Sociable
2724:Amicable
2535:Abundant
2515:dynamics
2337:Schröder
2327:Narayana
2297:Eulerian
2287:Delannoy
2282:Dedekind
2103:centered
1969:centered
1856:Amenable
1813:Narayana
1803:Leonardo
1699:Mersenne
1647:Powerful
1587:Achilles
1366:See also
95:= 4 and
3421:related
3385:related
3349:related
3347:Sorting
3232:Vampire
3217:Harshad
3159:related
3131:Repunit
3045:Lychrel
3020:Dudeney
2872:Størmer
2867:Sphenic
2852:Regular
2790:divisor
2729:Perfect
2625:Sublime
2595:Perfect
2322:Motzkin
2277:Catalan
1818:Padovan
1752:Leyland
1747:Idoneal
1742:Hilbert
1714:Woodall
1516:0225709
1486:in the
1483:A092697
1481::
1357:in the
1354:A128858
1315:in the
1312:A128857
1217:131ᘔ8ᘔ
1105:Digits
972:142857
958:102564
905:Digits
883:in the
880:A092697
40:of its
3287:Odious
3212:Frugal
3166:Cyclic
3155:Digit-
2862:Smooth
2847:Pronic
2807:Cyclic
2784:Other
2757:Euclid
2407:Wilson
2381:Primes
2040:Square
1909:Polite
1871:Riesel
1866:Knödel
1828:Perrin
1709:Thabit
1694:Fermat
1684:Cullen
1607:Square
1575:Powers
1522:
1514:
1343:− 1).
1067:Other
450:modulo
448:of 10
440:where
246:179487
227:717948
198:179487
155:717948
3328:Prime
3323:Lucky
3312:sieve
3241:Other
3227:Smith
3107:Digit
3065:Happy
3040:Keith
3013:Other
2857:Rough
2827:Giuga
2292:Euler
2154:Cubic
1808:Lucas
1704:Proth
1520:S2CID
1383:Notes
1277:Ɛ/ᘔƐ
1268:(11)
1248:(10)
1240:9/8Ɛ
1209:7/6Ɛ
1195:6/5Ɛ
1181:5/4Ɛ
1167:4/3Ɛ
1144:2497
1136:2/1Ɛ
1037:9/89
1023:8/79
1009:7/69
995:6/59
964:4/39
950:3/29
936:2/19
275:Thus
151:17948
145:17948
143:7 = 3
133:7 = 3
123:7 = 1
113:7 = 3
38:digit
3282:Evil
2962:Self
2912:and
2802:Blum
2513:and
2317:Lobb
2272:Cake
2267:Bell
2017:Star
1924:Ulam
1823:Pell
1612:Cube
1479:OEIS
1359:OEIS
1339:/(10
1327:/(10
1317:OEIS
1122:1/Ɛ
922:1/9
885:OEIS
457:− 1)
376:/(10
153:7 =
141:7948
135:7948
99:= 7
3400:Ban
2788:or
2307:Lah
1504:doi
1500:105
1274:55
1254:14
1237:45
1206:35
1192:2Ɛ
1178:25
1164:1Ɛ
1079:In
1034:44
1020:13
1006:22
992:58
947:28
933:18
453:(10
131:948
125:948
79:An
69:not
32:by
16:An
3470::
1518:,
1512:MR
1510:,
1498:,
1465:,
1442:,
1421:,
1400:,
1258:12
1231:9
1220:6
1214:8
1200:7
1186:6
1172:5
1158:4
1147:4
1141:3
1133:Ɛ
1127:2
1119:1
1116:1
1113:1
1060:≥
1028:9
1014:8
1000:7
986:6
975:6
969:5
961:6
955:4
941:3
927:2
919:1
916:1
913:1
887:)
546:0.
538:19
494:0.
486:19
459:.
413:10
394:10
364:≥
331:39
308:10
255:10
241:7.
222:0.
193:0.
121:48
115:48
1672:a
1553:e
1546:t
1539:v
1506::
1490:.
1449:.
1428:.
1407:.
1361:)
1341:n
1337:n
1335:-
1329:n
1325:n
1319:)
1302:m
1298:n
1296:/
1294:m
1290:m
1266:Ɛ
1246:ᘔ
1224:ᘔ
1151:7
1100:n
1094:n
1085:n
1069:n
1062:n
1058:k
1054:n
1050:n
979:7
900:n
894:n
867:n
851:n
804:k
800:n
604:n
600:n
596:n
576:m
559:.
543:=
535:2
507:.
491:=
483:1
468:n
464:n
455:n
442:m
428:)
425:1
417:m
409:(
403:1
397:n
390:k
378:n
374:k
366:n
362:k
357:k
353:n
336:.
328:7
323:=
320:x
304:x
301:+
298:7
292:=
289:x
286:4
260:.
235:=
219:=
216:x
213:4
190:=
181:=
178:x
157:.
111:8
105:8
97:k
93:n
89:n
85:k
81:n
46:n
34:n
20:-
18:n
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