3460:
869:
605:
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
604:
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
808:
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
281:
1319:
1, 105263157894736842, 1034482758620689655172413793, 102564, 102040816326530612244897959183673469387755, 1016949152542372881355932203389830508474576271186440677966, 1014492753623188405797, 1012658227848, 10112359550561797752808988764044943820224719, 10,
817: = 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.
1361:
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
184:
580:
528:
357:
449:
1562:
1320:
100917431192660550458715596330275229357798165137614678899082568807339449541284403669724770642201834862385321, 100840336134453781512605042016806722689075630252, ... (sequence
609:-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
3484:
1498:
1489:
1369:
1327:
895:
1555:
1082:-parasitic integers can be built by concatenation. For example, since 179487 is a 4-parasitic number, so are 179487179487, 179487179487179487 etc.
276:{\displaystyle x=0.179487179487179487\ldots =0.{\overline {179487}}{\mbox{ has }}4x=0.{\overline {717948}}={\frac {7.{\overline {179487}}}{10}}.}
539:
487:
292:
2362:
1548:
2357:
2372:
2352:
3065:
2645:
613:-parasitic number has not been found, is simply to not shift the product from the multiplication, but use it as is, and append
596:
105263157894736842 × 2 = 210526315789473684, which is the result of moving the last digit of 105263157894736842 to the front.
2367:
1387:
3151:
2467:
2817:
2136:
1929:
1098:-parasitic numbers are: (using inverted two and three for ten and eleven, respectively) (leading zeros are not allowed)
2852:
2822:
2497:
2487:
1429:
2993:
2407:
2141:
2121:
2683:
2847:
2942:
2565:
2322:
2131:
2113:
2007:
1997:
1987:
2827:
394:
3070:
2615:
2236:
2022:
2017:
2012:
2002:
1979:
2055:
59:
is itself a single-digit positive natural number. In other words, the decimal representation undergoes a right
2312:
3181:
3146:
2932:
2842:
2716:
2691:
2600:
2590:
2202:
2184:
2104:
1481:
1413:
3441:
2711:
2585:
2216:
1992:
1772:
1699:
1450:
2696:
2550:
2477:
1632:
172:
So 179487 is a 4-parasitic number with units digit 7. Others are 179487179487, 179487179487179487, etc.
3405:
3045:
3338:
3232:
3196:
2937:
2660:
2640:
2457:
2126:
1914:
1886:
1473:
456:
3060:
2924:
2919:
2887:
2650:
2625:
2620:
2595:
2525:
2521:
2452:
2342:
2174:
1970:
1939:
3459:
3463:
3217:
3212:
3126:
3100:
2998:
2977:
2749:
2630:
2580:
2502:
2472:
2412:
2179:
2159:
2090:
1803:
1530:
460:
102:
or greater) in the rightmost (units) place, and working up one digit at a time. For example, for
2347:
3357:
3302:
3156:
3131:
3105:
2882:
2560:
2555:
2482:
2462:
2447:
2169:
2151:
2070:
2060:
2045:
1823:
1808:
381:
176:
1408:
3393:
2772:
2744:
2734:
2726:
2610:
2575:
2570:
2537:
2231:
2194:
2085:
2080:
2075:
2065:
2037:
1924:
1876:
1871:
1828:
1767:
1514:
1526:
3369:
3258:
3191:
3117:
3040:
3014:
2832:
2545:
2402:
2337:
2307:
2297:
2292:
1958:
1866:
1813:
1657:
1597:
1522:
48:
617:(in this case 5) to the end. After 42 steps, the proper parasitic number will be found.
3374:
3242:
3227:
3091:
3055:
3030:
2906:
2877:
2862:
2739:
2635:
2605:
2332:
2287:
2164:
1762:
1757:
1752:
1724:
1709:
1622:
1607:
1585:
1572:
1455:
1434:
1343:
60:
40:
36:
3478:
3297:
3281:
3222:
3176:
2872:
2857:
2767:
2492:
2050:
1919:
1881:
1838:
1719:
1704:
1694:
1652:
1642:
1617:
1534:
1382:
885:
809:
remove leading zeros. Look at Table Three below displaying the derivation steps for
3333:
3322:
3237:
3075:
3050:
2967:
2867:
2837:
2812:
2796:
2701:
2668:
2417:
2391:
2302:
2241:
1818:
1714:
1647:
1627:
1602:
72:
3292:
3167:
2972:
2436:
2327:
2282:
2277:
2027:
1934:
1833:
1662:
1637:
1612:
3429:
3410:
2706:
2317:
1276:
1256:
1091:
1540:
1075:
then the numbers do not start with zero and hence fit the actual definition.
3035:
2962:
2954:
2759:
2673:
1791:
1059:
In general, if we relax the rules to allow a leading zero, then there are 9
802:
17. 5 × 59183673469387755 = 295918367346938775 − Shift = 959183673469387755
17:
868:
3136:
797:
16. 5 × 9183673469387755 = 45918367346938775 − Shift = 59183673469387755
3141:
2800:
1518:
52:
792:
15. 5 × 183673469387755 = 918367346938775 − Shift = 9183673469387755
366:-parasitic number can be found as follows. Pick a one digit integer
1505:
Bernstein, Leon (1968), "Multiplicative twins and primitive roots",
708:
17. 5 × 183673469387755 = 918367346938775 − Shift = 183673469387755
703:
16. 5 × 183673469387755 = 918367346938775 − Shift = 183673469387755
698:
15. 5 × 183673469387755 = 918367346938775 − Shift = 183673469387755
787:
14. 5 × 83673469387755 = 418367346938775 − Shift = 183673469387755
693:
14. 5 × 83673469387755 = 418367346938775 − Shift = 183673469387755
575:{\displaystyle {\frac {2}{19}}=0.{\overline {105263157894736842}}.}
523:{\displaystyle {\frac {1}{19}}=0.{\overline {052631578947368421}}.}
352:{\displaystyle 4x={\frac {7+x}{10}}{\mbox{ so }}x={\frac {7}{39}}.}
1282:
1011235930336ᘔ53909ᘔ873Ɛ325819Ɛ9975055Ɛ54ᘔ3145ᘔ42694157078404491Ɛ
867:
78:
So even though 4 × 25641 = 102564, the number 25641 is
782:
13. 5 × 3673469387755 = 18367346938775 − Shift = 83673469387755
688:
13. 5 × 3673469387755 = 18367346938775 − Shift = 83673469387755
589:
of this period is 18, the same as the order of 10 modulo 19, so
3427:
3391:
3355:
3319:
3279:
2904:
2793:
2519:
2434:
2389:
2266:
1956:
1903:
1855:
1789:
1741:
1679:
1583:
1544:
777:
12. 5 × 673469387755 = 3367346938775 − Shift = 3673469387755
683:
12. 5 × 673469387755 = 3367346938775 − Shift = 3673469387755
1493:
1364:
1322:
1000:
1016949152542372881355932203389830508474576271186440677966
890:
772:
11. 5 × 73469387755 = 367346938775 − Shift = 673469387755
678:
11. 5 × 73469387755 = 367346938775 − Shift = 673469387755
94:-parasitic number can be derived by starting with a digit
75:
to be used, and that is a commonly followed convention.
67:
4 × 128205 = 512820, so 128205 is 4-parasitic.
1311:
is obtained merely by shifting the leftmost digit 1 of
767:
10. 5 × 3469387755 = 17346938775 − Shift = 73469387755
673:
10. 5 × 3469387755 = 17346938775 − Shift = 73469387755
1245:
101419648634459Ɛ9384Ɛ26Ɛ533040547216ᘔ1155Ɛ3Ɛ12978ᘔ399
888:. They are: (leading zeros are not allowed) (sequence
324:
217:
542:
490:
397:
295:
187:
3251:
3205:
3165:
3116:
3090:
3023:
3007:
2986:
2953:
2918:
2758:
2725:
2682:
2659:
2536:
2224:
2215:
2193:
2150:
2112:
2103:
2036:
1978:
1969:
884:, after a puzzle concerning these numbers posed by
762:9. 5 × 469387755 = 2346938775 − Shift = 3469387755
668:9. 5 × 469387755 = 2346938775 − Shift = 3469387755
574:
522:
443:
351:
275:
757:8. 5 × 69387755 = 346938775 − Shift = 469387755
663:8. 5 × 69387755 = 346938775 − Shift = 469387755
481:− 1 = 19 and the repeating decimal for 1/19 is
1556:
1042:10112359550561797752808988764044943820224719
752:7. 5 × 9387755 = 46938775 − Shift = 69387755
658:7. 5 × 9387755 = 46938775 − Shift = 69387755
8:
444:{\displaystyle {\frac {k}{10n-1}}(10^{m}-1)}
3424:
3388:
3352:
3316:
3276:
2950:
2915:
2901:
2790:
2533:
2516:
2431:
2386:
2263:
2221:
2109:
1975:
1966:
1953:
1900:
1857:Possessing a specific set of other numbers
1852:
1786:
1738:
1676:
1580:
1563:
1549:
1541:
1214:101899Ɛ864406Ɛ33ᘔᘔ15423913745949305255Ɛ17
747:6. 5 × 387755 = 1938775 − Shift = 9387755
653:6. 5 × 387755 = 1938775 − Shift = 9387755
1499:On-Line Encyclopedia of Integer Sequences
559:
543:
541:
507:
491:
489:
426:
398:
396:
336:
323:
305:
294:
254:
248:
235:
216:
206:
186:
1303:beginning with 1 such that the quotient
1100:
900:
853:6. 4 × 102564 = 410256 − Shift = 102564
824:
718:
624:
1430:"Freeman Dyson's 4th-Grade Math Puzzle"
1399:
848:5. 4 × 02564 = 010256 − Shift = 102564
742:5. 5 × 87755 = 438775 − Shift = 387755
648:5. 5 × 87755 = 438775 − Shift = 387755
455:is the length of the period; i.e. the
1407:Dawidoff, Nicholas (March 25, 2009),
880:-parasitic numbers are also known as
7:
1200:1020408142854ᘔ997732650ᘔ18346916306
843:4. 4 × 2564 = 10256 − Shift = 02564
737:4. 5 × 7755 = 38775 − Shift = 87755
643:4. 5 × 7755 = 38775 − Shift = 87755
1299:In strict definition, least number
47:, results in movement of the last
25:
838:3. 4 × 564 = 2256 − Shift = 2564
732:3. 5 × 755 = 3775 − Shift = 7755
638:3. 5 × 755 = 3775 − Shift = 7755
533:So that for 2/19 is double that:
71:Most mathematicians do not allow
3485:Base-dependent integer sequences
3458:
3066:Perfect digit-to-digit invariant
1449:Tierney, John (April 13, 2009),
1428:Tierney, John (April 6, 2009),
1388:Linear-feedback shift register
1186:1025355ᘔ9433073ᘔ458409919Ɛ715
833:2. 4 × 64 = 256 − Shift = 564
727:2. 5 × 55 = 275 − Shift = 755
633:2. 5 × 55 = 275 − Shift = 755
438:
419:
1:
1905:Expressible via specific sums
1357:Number of digits of them are
1342:− 1), also the period of the
955:1034482758620689655172413793
380:, and take the period of the
63:by one place. For example:
1063:-parasitic numbers for each
564:
512:
259:
240:
211:
2994:Multiplicative digital root
828:1. 4 × 4 = 16 − Shift = 64
722:1. 5 × 5 = 25 − Shift = 55
628:1. 5 × 5 = 25 − Shift = 55
35:(in base 10) is a positive
3501:
98:(which should be equal to
3454:
3437:
3423:
3401:
3387:
3365:
3351:
3329:
3315:
3288:
3275:
3071:Perfect digital invariant
2914:
2900:
2808:
2789:
2646:Superior highly composite
2532:
2515:
2443:
2430:
2398:
2385:
2273:
2262:
1965:
1952:
1910:
1899:
1862:
1851:
1799:
1785:
1748:
1737:
1690:
1675:
1593:
1579:
1507:Mathematische Zeitschrift
2684:Euler's totient function
2468:Euler–Jacobi pseudoprime
1743:Other polynomial numbers
1451:"Prize for Dyson Puzzle"
1172:10309236ᘔ88206164719544
473:For another example, if
2498:Somer–Lucas pseudoprime
2488:Lucas–Carmichael number
2323:Lazy caterer's sequence
1482:Oxford University Press
1414:New York Times Magazine
1334:They are the period of
1014:1014492753623188405797
2373:Wedderburn–Etherington
1773:Lucky numbers of Euler
873:
600:Additional information
593:= 105263157894736842.
576:
524:
445:
353:
277:
53:decimal representation
2661:Prime omega functions
2478:Frobenius pseudoprime
2268:Combinatorial numbers
2137:Centered dodecahedral
1930:Primary pseudoperfect
1315:to the right end are
1094:system, the smallest
872:Freeman Dyson in 2005
871:
577:
525:
446:
354:
278:
3120:-composition related
2920:Arithmetic functions
2522:Arithmetic functions
2458:Elliptic pseudoprime
2142:Centered icosahedral
2122:Centered tetrahedral
1067:. Otherwise only if
540:
488:
457:multiplicative order
395:
293:
195:0.179487179487179487
185:
3046:Kaprekar's constant
2566:Colossally abundant
2453:Catalan pseudoprime
2353:Schröder–Hipparchus
2132:Centered octahedral
2008:Centered heptagonal
1998:Centered pentagonal
1988:Centered triangular
1588:and related numbers
1409:"The Civil Heretic"
941:105263157894736842
813: = 4 and
114:4 × 7 = 2
55:to its front. Here
3464:Mathematics portal
3406:Aronson's sequence
3152:Smarandache–Wellin
2909:-dependent numbers
2616:Primitive abundant
2503:Strong pseudoprime
2493:Perrin pseudoprime
2473:Fermat pseudoprime
2413:Wolstenholme prime
2237:Squared triangular
2023:Centered decagonal
2018:Centered nonagonal
2013:Centered octagonal
2003:Centered hexagonal
1519:10.1007/BF01135448
1478:Wonders of Numbers
1113:-parasitic number
913:-parasitic number
874:
864:-parasitic numbers
572:
562:105263157894736842
520:
510:052631578947368421
441:
391:−1). This will be
349:
328:
273:
221:
3472:
3471:
3450:
3449:
3419:
3418:
3383:
3382:
3347:
3346:
3311:
3310:
3271:
3270:
3267:
3266:
3086:
3085:
2896:
2895:
2785:
2784:
2781:
2780:
2727:Aliquot sequences
2538:Divisor functions
2511:
2510:
2483:Lucas pseudoprime
2463:Euler pseudoprime
2448:Carmichael number
2426:
2425:
2381:
2380:
2258:
2257:
2254:
2253:
2250:
2249:
2211:
2210:
2099:
2098:
2056:Square triangular
1948:
1947:
1895:
1894:
1847:
1846:
1781:
1780:
1733:
1732:
1671:
1670:
1295:Strict definition
1292:
1291:
1262:14Ɛ36429ᘔ7085792
1052:
1051:
857:
856:
806:
805:
712:
711:
567:
551:
515:
499:
417:
382:repeating decimal
344:
327:
321:
268:
262:
243:
220:
214:
177:repeating decimal
16:(Redirected from
3492:
3462:
3425:
3394:Natural language
3389:
3353:
3321:Generated via a
3317:
3277:
3182:Digit-reassembly
3147:Self-descriptive
2951:
2916:
2902:
2853:Lucas–Carmichael
2843:Harmonic divisor
2791:
2717:Sparsely totient
2692:Highly cototient
2601:Multiply perfect
2591:Highly composite
2534:
2517:
2432:
2387:
2368:Telephone number
2264:
2222:
2203:Square pyramidal
2185:Stella octangula
2110:
1976:
1967:
1959:Figurate numbers
1954:
1901:
1853:
1787:
1739:
1677:
1581:
1565:
1558:
1551:
1542:
1537:
1496:
1461:
1459:
1446:
1440:
1438:
1425:
1419:
1417:
1404:
1367:
1325:
1101:
901:
893:
825:
719:
625:
592:
581:
579:
578:
573:
568:
560:
552:
544:
529:
527:
526:
521:
516:
508:
500:
492:
469:
450:
448:
447:
442:
431:
430:
418:
416:
399:
379:
358:
356:
355:
350:
345:
337:
329:
325:
322:
317:
306:
282:
280:
279:
274:
269:
264:
263:
255:
249:
244:
236:
222:
218:
215:
207:
175:Notice that the
33:parasitic number
21:
3500:
3499:
3495:
3494:
3493:
3491:
3490:
3489:
3475:
3474:
3473:
3468:
3446:
3442:Strobogrammatic
3433:
3415:
3397:
3379:
3361:
3343:
3325:
3307:
3284:
3263:
3247:
3206:Divisor-related
3201:
3161:
3112:
3082:
3019:
3003:
2982:
2949:
2922:
2910:
2892:
2804:
2803:related numbers
2777:
2754:
2721:
2712:Perfect totient
2678:
2655:
2586:Highly abundant
2528:
2507:
2439:
2422:
2394:
2377:
2363:Stirling second
2269:
2246:
2207:
2189:
2146:
2095:
2032:
1993:Centered square
1961:
1944:
1906:
1891:
1858:
1843:
1795:
1794:defined numbers
1777:
1744:
1729:
1700:Double Mersenne
1686:
1667:
1589:
1575:
1573:natural numbers
1569:
1504:
1488:
1470:
1465:
1464:
1448:
1447:
1443:
1427:
1426:
1422:
1406:
1405:
1401:
1396:
1379:
1363:
1344:decadic integer
1321:
1297:
1088:
1057:
889:
866:
823:
717:
623:
602:
591:2 × (10 − 1)/19
590:
538:
537:
486:
485:
463:
422:
403:
393:
392:
371:
362:In general, an
307:
291:
290:
250:
219: has
183:
182:
88:
23:
22:
15:
12:
11:
5:
3498:
3496:
3488:
3487:
3477:
3476:
3470:
3469:
3467:
3466:
3455:
3452:
3451:
3448:
3447:
3445:
3444:
3438:
3435:
3434:
3428:
3421:
3420:
3417:
3416:
3414:
3413:
3408:
3402:
3399:
3398:
3392:
3385:
3384:
3381:
3380:
3378:
3377:
3375:Sorting number
3372:
3370:Pancake number
3366:
3363:
3362:
3356:
3349:
3348:
3345:
3344:
3342:
3341:
3336:
3330:
3327:
3326:
3320:
3313:
3312:
3309:
3308:
3306:
3305:
3300:
3295:
3289:
3286:
3285:
3282:Binary numbers
3280:
3273:
3272:
3269:
3268:
3265:
3264:
3262:
3261:
3255:
3253:
3249:
3248:
3246:
3245:
3240:
3235:
3230:
3225:
3220:
3215:
3209:
3207:
3203:
3202:
3200:
3199:
3194:
3189:
3184:
3179:
3173:
3171:
3163:
3162:
3160:
3159:
3154:
3149:
3144:
3139:
3134:
3129:
3123:
3121:
3114:
3113:
3111:
3110:
3109:
3108:
3097:
3095:
3092:P-adic numbers
3088:
3087:
3084:
3083:
3081:
3080:
3079:
3078:
3068:
3063:
3058:
3053:
3048:
3043:
3038:
3033:
3027:
3025:
3021:
3020:
3018:
3017:
3011:
3009:
3008:Coding-related
3005:
3004:
3002:
3001:
2996:
2990:
2988:
2984:
2983:
2981:
2980:
2975:
2970:
2965:
2959:
2957:
2948:
2947:
2946:
2945:
2943:Multiplicative
2940:
2929:
2927:
2912:
2911:
2907:Numeral system
2905:
2898:
2897:
2894:
2893:
2891:
2890:
2885:
2880:
2875:
2870:
2865:
2860:
2855:
2850:
2845:
2840:
2835:
2830:
2825:
2820:
2815:
2809:
2806:
2805:
2794:
2787:
2786:
2783:
2782:
2779:
2778:
2776:
2775:
2770:
2764:
2762:
2756:
2755:
2753:
2752:
2747:
2742:
2737:
2731:
2729:
2723:
2722:
2720:
2719:
2714:
2709:
2704:
2699:
2697:Highly totient
2694:
2688:
2686:
2680:
2679:
2677:
2676:
2671:
2665:
2663:
2657:
2656:
2654:
2653:
2648:
2643:
2638:
2633:
2628:
2623:
2618:
2613:
2608:
2603:
2598:
2593:
2588:
2583:
2578:
2573:
2568:
2563:
2558:
2553:
2551:Almost perfect
2548:
2542:
2540:
2530:
2529:
2520:
2513:
2512:
2509:
2508:
2506:
2505:
2500:
2495:
2490:
2485:
2480:
2475:
2470:
2465:
2460:
2455:
2450:
2444:
2441:
2440:
2435:
2428:
2427:
2424:
2423:
2421:
2420:
2415:
2410:
2405:
2399:
2396:
2395:
2390:
2383:
2382:
2379:
2378:
2376:
2375:
2370:
2365:
2360:
2358:Stirling first
2355:
2350:
2345:
2340:
2335:
2330:
2325:
2320:
2315:
2310:
2305:
2300:
2295:
2290:
2285:
2280:
2274:
2271:
2270:
2267:
2260:
2259:
2256:
2255:
2252:
2251:
2248:
2247:
2245:
2244:
2239:
2234:
2228:
2226:
2219:
2213:
2212:
2209:
2208:
2206:
2205:
2199:
2197:
2191:
2190:
2188:
2187:
2182:
2177:
2172:
2167:
2162:
2156:
2154:
2148:
2147:
2145:
2144:
2139:
2134:
2129:
2124:
2118:
2116:
2107:
2101:
2100:
2097:
2096:
2094:
2093:
2088:
2083:
2078:
2073:
2068:
2063:
2058:
2053:
2048:
2042:
2040:
2034:
2033:
2031:
2030:
2025:
2020:
2015:
2010:
2005:
2000:
1995:
1990:
1984:
1982:
1973:
1963:
1962:
1957:
1950:
1949:
1946:
1945:
1943:
1942:
1937:
1932:
1927:
1922:
1917:
1911:
1908:
1907:
1904:
1897:
1896:
1893:
1892:
1890:
1889:
1884:
1879:
1874:
1869:
1863:
1860:
1859:
1856:
1849:
1848:
1845:
1844:
1842:
1841:
1836:
1831:
1826:
1821:
1816:
1811:
1806:
1800:
1797:
1796:
1790:
1783:
1782:
1779:
1778:
1776:
1775:
1770:
1765:
1760:
1755:
1749:
1746:
1745:
1742:
1735:
1734:
1731:
1730:
1728:
1727:
1722:
1717:
1712:
1707:
1702:
1697:
1691:
1688:
1687:
1680:
1673:
1672:
1669:
1668:
1666:
1665:
1660:
1655:
1650:
1645:
1640:
1635:
1630:
1625:
1620:
1615:
1610:
1605:
1600:
1594:
1591:
1590:
1584:
1577:
1576:
1570:
1568:
1567:
1560:
1553:
1545:
1539:
1538:
1502:
1485:
1480:, Chapter 28,
1474:C. A. Pickover
1469:
1466:
1463:
1462:
1456:New York Times
1441:
1435:New York Times
1420:
1398:
1397:
1395:
1392:
1391:
1390:
1385:
1378:
1375:
1374:
1373:
1332:
1331:
1296:
1293:
1290:
1289:
1286:
1283:
1280:
1273:
1272:
1266:
1263:
1260:
1253:
1252:
1249:
1246:
1243:
1239:
1238:
1232:
1229:
1226:
1222:
1221:
1218:
1215:
1212:
1208:
1207:
1204:
1201:
1198:
1194:
1193:
1190:
1187:
1184:
1180:
1179:
1176:
1173:
1170:
1166:
1165:
1159:
1156:
1153:
1149:
1148:
1145:
1142:
1139:
1135:
1134:
1131:
1128:
1125:
1121:
1120:
1117:
1114:
1107:
1087:
1084:
1056:
1053:
1050:
1049:
1046:
1043:
1040:
1036:
1035:
1032:
1029:
1028:1012658227848
1026:
1022:
1021:
1018:
1015:
1012:
1008:
1007:
1004:
1001:
998:
994:
993:
987:
984:
981:
977:
976:
973:
970:
967:
963:
962:
959:
956:
953:
949:
948:
945:
942:
939:
935:
934:
931:
928:
925:
921:
920:
917:
914:
907:
865:
858:
855:
854:
850:
849:
845:
844:
840:
839:
835:
834:
830:
829:
822:
819:
804:
803:
799:
798:
794:
793:
789:
788:
784:
783:
779:
778:
774:
773:
769:
768:
764:
763:
759:
758:
754:
753:
749:
748:
744:
743:
739:
738:
734:
733:
729:
728:
724:
723:
716:
713:
710:
709:
705:
704:
700:
699:
695:
694:
690:
689:
685:
684:
680:
679:
675:
674:
670:
669:
665:
664:
660:
659:
655:
654:
650:
649:
645:
644:
640:
639:
635:
634:
630:
629:
622:
619:
601:
598:
583:
582:
571:
566:
563:
558:
555:
550:
547:
531:
530:
519:
514:
511:
506:
503:
498:
495:
440:
437:
434:
429:
425:
421:
415:
412:
409:
406:
402:
360:
359:
348:
343:
340:
335:
332:
326: so
320:
316:
313:
310:
304:
301:
298:
284:
283:
272:
267:
261:
258:
253:
247:
242:
239:
234:
231:
228:
225:
213:
210:
205:
202:
199:
196:
193:
190:
170:
169:
160:4 ×
158:
150:4 ×
148:
140:4 ×
138:
130:4 ×
128:
120:4 ×
118:
87:
84:
69:
68:
61:circular shift
37:natural number
24:
14:
13:
10:
9:
6:
4:
3:
2:
3497:
3486:
3483:
3482:
3480:
3465:
3461:
3457:
3456:
3453:
3443:
3440:
3439:
3436:
3431:
3426:
3422:
3412:
3409:
3407:
3404:
3403:
3400:
3395:
3390:
3386:
3376:
3373:
3371:
3368:
3367:
3364:
3359:
3354:
3350:
3340:
3337:
3335:
3332:
3331:
3328:
3324:
3318:
3314:
3304:
3301:
3299:
3296:
3294:
3291:
3290:
3287:
3283:
3278:
3274:
3260:
3257:
3256:
3254:
3250:
3244:
3241:
3239:
3236:
3234:
3233:Polydivisible
3231:
3229:
3226:
3224:
3221:
3219:
3216:
3214:
3211:
3210:
3208:
3204:
3198:
3195:
3193:
3190:
3188:
3185:
3183:
3180:
3178:
3175:
3174:
3172:
3169:
3164:
3158:
3155:
3153:
3150:
3148:
3145:
3143:
3140:
3138:
3135:
3133:
3130:
3128:
3125:
3124:
3122:
3119:
3115:
3107:
3104:
3103:
3102:
3099:
3098:
3096:
3093:
3089:
3077:
3074:
3073:
3072:
3069:
3067:
3064:
3062:
3059:
3057:
3054:
3052:
3049:
3047:
3044:
3042:
3039:
3037:
3034:
3032:
3029:
3028:
3026:
3022:
3016:
3013:
3012:
3010:
3006:
3000:
2997:
2995:
2992:
2991:
2989:
2987:Digit product
2985:
2979:
2976:
2974:
2971:
2969:
2966:
2964:
2961:
2960:
2958:
2956:
2952:
2944:
2941:
2939:
2936:
2935:
2934:
2931:
2930:
2928:
2926:
2921:
2917:
2913:
2908:
2903:
2899:
2889:
2886:
2884:
2881:
2879:
2876:
2874:
2871:
2869:
2866:
2864:
2861:
2859:
2856:
2854:
2851:
2849:
2846:
2844:
2841:
2839:
2836:
2834:
2831:
2829:
2826:
2824:
2823:Erdős–Nicolas
2821:
2819:
2816:
2814:
2811:
2810:
2807:
2802:
2798:
2792:
2788:
2774:
2771:
2769:
2766:
2765:
2763:
2761:
2757:
2751:
2748:
2746:
2743:
2741:
2738:
2736:
2733:
2732:
2730:
2728:
2724:
2718:
2715:
2713:
2710:
2708:
2705:
2703:
2700:
2698:
2695:
2693:
2690:
2689:
2687:
2685:
2681:
2675:
2672:
2670:
2667:
2666:
2664:
2662:
2658:
2652:
2649:
2647:
2644:
2642:
2641:Superabundant
2639:
2637:
2634:
2632:
2629:
2627:
2624:
2622:
2619:
2617:
2614:
2612:
2609:
2607:
2604:
2602:
2599:
2597:
2594:
2592:
2589:
2587:
2584:
2582:
2579:
2577:
2574:
2572:
2569:
2567:
2564:
2562:
2559:
2557:
2554:
2552:
2549:
2547:
2544:
2543:
2541:
2539:
2535:
2531:
2527:
2523:
2518:
2514:
2504:
2501:
2499:
2496:
2494:
2491:
2489:
2486:
2484:
2481:
2479:
2476:
2474:
2471:
2469:
2466:
2464:
2461:
2459:
2456:
2454:
2451:
2449:
2446:
2445:
2442:
2438:
2433:
2429:
2419:
2416:
2414:
2411:
2409:
2406:
2404:
2401:
2400:
2397:
2393:
2388:
2384:
2374:
2371:
2369:
2366:
2364:
2361:
2359:
2356:
2354:
2351:
2349:
2346:
2344:
2341:
2339:
2336:
2334:
2331:
2329:
2326:
2324:
2321:
2319:
2316:
2314:
2311:
2309:
2306:
2304:
2301:
2299:
2296:
2294:
2291:
2289:
2286:
2284:
2281:
2279:
2276:
2275:
2272:
2265:
2261:
2243:
2240:
2238:
2235:
2233:
2230:
2229:
2227:
2223:
2220:
2218:
2217:4-dimensional
2214:
2204:
2201:
2200:
2198:
2196:
2192:
2186:
2183:
2181:
2178:
2176:
2173:
2171:
2168:
2166:
2163:
2161:
2158:
2157:
2155:
2153:
2149:
2143:
2140:
2138:
2135:
2133:
2130:
2128:
2127:Centered cube
2125:
2123:
2120:
2119:
2117:
2115:
2111:
2108:
2106:
2105:3-dimensional
2102:
2092:
2089:
2087:
2084:
2082:
2079:
2077:
2074:
2072:
2069:
2067:
2064:
2062:
2059:
2057:
2054:
2052:
2049:
2047:
2044:
2043:
2041:
2039:
2035:
2029:
2026:
2024:
2021:
2019:
2016:
2014:
2011:
2009:
2006:
2004:
2001:
1999:
1996:
1994:
1991:
1989:
1986:
1985:
1983:
1981:
1977:
1974:
1972:
1971:2-dimensional
1968:
1964:
1960:
1955:
1951:
1941:
1938:
1936:
1933:
1931:
1928:
1926:
1923:
1921:
1918:
1916:
1915:Nonhypotenuse
1913:
1912:
1909:
1902:
1898:
1888:
1885:
1883:
1880:
1878:
1875:
1873:
1870:
1868:
1865:
1864:
1861:
1854:
1850:
1840:
1837:
1835:
1832:
1830:
1827:
1825:
1822:
1820:
1817:
1815:
1812:
1810:
1807:
1805:
1802:
1801:
1798:
1793:
1788:
1784:
1774:
1771:
1769:
1766:
1764:
1761:
1759:
1756:
1754:
1751:
1750:
1747:
1740:
1736:
1726:
1723:
1721:
1718:
1716:
1713:
1711:
1708:
1706:
1703:
1701:
1698:
1696:
1693:
1692:
1689:
1684:
1678:
1674:
1664:
1661:
1659:
1656:
1654:
1653:Perfect power
1651:
1649:
1646:
1644:
1643:Seventh power
1641:
1639:
1636:
1634:
1631:
1629:
1626:
1624:
1621:
1619:
1616:
1614:
1611:
1609:
1606:
1604:
1601:
1599:
1596:
1595:
1592:
1587:
1582:
1578:
1574:
1566:
1561:
1559:
1554:
1552:
1547:
1546:
1543:
1536:
1532:
1528:
1524:
1520:
1516:
1512:
1508:
1503:
1500:
1495:
1491:
1486:
1483:
1479:
1475:
1472:
1471:
1467:
1458:
1457:
1452:
1445:
1442:
1437:
1436:
1431:
1424:
1421:
1416:
1415:
1410:
1403:
1400:
1393:
1389:
1386:
1384:
1383:Cyclic number
1381:
1380:
1376:
1371:
1366:
1360:
1359:
1358:
1355:
1353:
1349:
1345:
1341:
1337:
1329:
1324:
1318:
1317:
1316:
1314:
1310:
1306:
1302:
1294:
1287:
1284:
1281:
1278:
1275:
1274:
1270:
1267:
1264:
1261:
1258:
1255:
1254:
1250:
1247:
1244:
1241:
1240:
1236:
1233:
1230:
1227:
1224:
1223:
1219:
1216:
1213:
1210:
1209:
1205:
1202:
1199:
1196:
1195:
1191:
1188:
1185:
1182:
1181:
1177:
1174:
1171:
1168:
1167:
1163:
1160:
1157:
1154:
1151:
1150:
1146:
1143:
1140:
1137:
1136:
1132:
1129:
1126:
1123:
1122:
1118:
1115:
1112:
1108:
1106:
1103:
1102:
1099:
1097:
1093:
1085:
1083:
1081:
1076:
1074:
1070:
1066:
1062:
1054:
1047:
1044:
1041:
1038:
1037:
1033:
1030:
1027:
1024:
1023:
1019:
1016:
1013:
1010:
1009:
1005:
1002:
999:
996:
995:
991:
988:
985:
982:
979:
978:
974:
971:
968:
965:
964:
960:
957:
954:
951:
950:
946:
943:
940:
937:
936:
932:
929:
926:
923:
922:
918:
915:
912:
908:
906:
903:
902:
899:
897:
892:
887:
886:Freeman Dyson
883:
882:Dyson numbers
879:
876:The smallest
870:
863:
859:
852:
851:
847:
846:
842:
841:
837:
836:
832:
831:
827:
826:
820:
818:
816:
812:
801:
800:
796:
795:
791:
790:
786:
785:
781:
780:
776:
775:
771:
770:
766:
765:
761:
760:
756:
755:
751:
750:
746:
745:
741:
740:
736:
735:
731:
730:
726:
725:
721:
720:
714:
707:
706:
702:
701:
697:
696:
692:
691:
687:
686:
682:
681:
677:
676:
672:
671:
667:
666:
662:
661:
657:
656:
652:
651:
647:
646:
642:
641:
637:
636:
632:
631:
627:
626:
620:
618:
616:
612:
608:
599:
597:
594:
588:
569:
561:
556:
553:
548:
545:
536:
535:
534:
517:
509:
504:
501:
496:
493:
484:
483:
482:
480:
476:
471:
467:
462:
458:
454:
435:
432:
427:
423:
413:
410:
407:
404:
400:
390:
386:
383:
378:
374:
369:
365:
346:
341:
338:
333:
330:
318:
314:
311:
308:
302:
299:
296:
289:
288:
287:
270:
265:
256:
251:
245:
237:
232:
229:
226:
223:
208:
203:
200:
197:
194:
191:
188:
181:
180:
179:
178:
173:
167:
163:
159:
157:
153:
149:
147:
143:
139:
137:
133:
129:
127:
123:
119:
117:
113:
112:
111:
109:
105:
101:
97:
93:
85:
83:
82:4-parasitic.
81:
76:
74:
73:leading zeros
66:
65:
64:
62:
58:
54:
50:
46:
42:
38:
34:
30:
19:
3197:Transposable
3186:
3061:Narcissistic
2968:Digital root
2888:Super-Poulet
2848:Jordan–Pólya
2797:prime factor
2702:Noncototient
2669:Almost prime
2651:Superperfect
2626:Refactorable
2621:Quasiperfect
2596:Hyperperfect
2437:Pseudoprimes
2408:Wall–Sun–Sun
2343:Ordered Bell
2313:Fuss–Catalan
2225:non-centered
2175:Dodecahedral
2152:non-centered
2038:non-centered
1940:Wolstenholme
1685:× 2 ± 1
1682:
1681:Of the form
1648:Eighth power
1628:Fourth power
1510:
1506:
1477:
1454:
1444:
1433:
1423:
1412:
1402:
1356:
1351:
1347:
1339:
1335:
1333:
1312:
1308:
1304:
1300:
1298:
1268:
1234:
1161:
1141:10631694842
1110:
1104:
1095:
1089:
1079:
1077:
1072:
1068:
1064:
1060:
1058:
1055:General note
989:
910:
904:
881:
877:
875:
861:
814:
810:
807:
614:
610:
606:
603:
595:
586:
584:
532:
478:
477:= 2, then 10
474:
472:
465:
452:
388:
384:
376:
372:
367:
363:
361:
285:
174:
171:
165:
161:
155:
151:
145:
141:
135:
131:
125:
121:
115:
107:
103:
99:
95:
91:
89:
79:
77:
70:
56:
44:
39:which, when
32:
28:
26:
18:Dyson number
3218:Extravagant
3213:Equidigital
3168:permutation
3127:Palindromic
3101:Automorphic
2999:Sum-product
2978:Sum-product
2933:Persistence
2828:Erdős–Woods
2750:Untouchable
2631:Semiperfect
2581:Hemiperfect
2242:Tesseractic
2180:Icosahedral
2160:Tetrahedral
2091:Dodecagonal
1792:Recursively
1663:Prime power
1638:Sixth power
1633:Fifth power
1613:Power of 10
1571:Classes of
1271:/9Ɛ = 2/15
1237:/7Ɛ = 2/17
1086:Other bases
821:Table Three
585:The length
3430:Graphemics
3303:Pernicious
3157:Undulating
3132:Pandigital
3106:Trimorphic
2707:Nontotient
2556:Arithmetic
2170:Octahedral
2071:Heptagonal
2061:Pentagonal
2046:Triangular
1887:Sierpiński
1809:Jacobsthal
1608:Power of 3
1603:Power of 2
1468:References
1164:/2Ɛ = 1/5
1119:Period of
1092:duodecimal
992:/49 = 1/7
919:Period of
370:such that
86:Derivation
41:multiplied
3187:Parasitic
3036:Factorion
2963:Digit sum
2955:Digit sum
2773:Fortunate
2760:Primorial
2674:Semiprime
2611:Practical
2576:Descartes
2571:Deficient
2561:Betrothed
2403:Wieferich
2232:Pentatope
2195:pyramidal
2086:Decagonal
2081:Nonagonal
2076:Octagonal
2066:Hexagonal
1925:Practical
1872:Congruent
1804:Fibonacci
1768:Loeschian
1535:121138247
1513:: 49–58,
1487:Sequence
1484:UK, 2000.
1109:Smallest
909:Smallest
860:Smallest
715:Table Two
621:Table One
565:¯
513:¯
433:−
411:−
260:¯
241:¯
212:¯
198:…
3479:Category
3259:Friedman
3192:Primeval
3137:Repdigit
3094:-related
3041:Kaprekar
3015:Meertens
2938:Additive
2925:dynamics
2833:Friendly
2745:Sociable
2735:Amicable
2546:Abundant
2526:dynamics
2348:Schröder
2338:Narayana
2308:Eulerian
2298:Delannoy
2293:Dedekind
2114:centered
1980:centered
1867:Amenable
1824:Narayana
1814:Leonardo
1710:Mersenne
1658:Powerful
1598:Achilles
1377:See also
106:= 4 and
3432:related
3396:related
3360:related
3358:Sorting
3243:Vampire
3228:Harshad
3170:related
3142:Repunit
3056:Lychrel
3031:Dudeney
2883:Størmer
2878:Sphenic
2863:Regular
2801:divisor
2740:Perfect
2636:Sublime
2606:Perfect
2333:Motzkin
2288:Catalan
1829:Padovan
1763:Leyland
1758:Idoneal
1753:Hilbert
1725:Woodall
1527:0225709
1497:in the
1494:A092697
1492::
1368:in the
1365:A128858
1326:in the
1323:A128857
1228:131ᘔ8ᘔ
1116:Digits
983:142857
969:102564
916:Digits
894:in the
891:A092697
51:of its
3298:Odious
3223:Frugal
3177:Cyclic
3166:Digit-
2873:Smooth
2858:Pronic
2818:Cyclic
2795:Other
2768:Euclid
2418:Wilson
2392:Primes
2051:Square
1920:Polite
1882:Riesel
1877:Knödel
1839:Perrin
1720:Thabit
1705:Fermat
1695:Cullen
1618:Square
1586:Powers
1533:
1525:
1354:− 1).
1078:Other
461:modulo
459:of 10
451:where
257:179487
238:717948
209:179487
166:717948
3339:Prime
3334:Lucky
3323:sieve
3252:Other
3238:Smith
3118:Digit
3076:Happy
3051:Keith
3024:Other
2868:Rough
2838:Giuga
2303:Euler
2165:Cubic
1819:Lucas
1715:Proth
1531:S2CID
1394:Notes
1288:Ɛ/ᘔƐ
1279:(11)
1259:(10)
1251:9/8Ɛ
1220:7/6Ɛ
1206:6/5Ɛ
1192:5/4Ɛ
1178:4/3Ɛ
1155:2497
1147:2/1Ɛ
1048:9/89
1034:8/79
1020:7/69
1006:6/59
975:4/39
961:3/29
947:2/19
286:Thus
162:17948
156:17948
154:7 = 3
144:7 = 3
134:7 = 1
124:7 = 3
49:digit
3293:Evil
2973:Self
2923:and
2813:Blum
2524:and
2328:Lobb
2283:Cake
2278:Bell
2028:Star
1935:Ulam
1834:Pell
1623:Cube
1490:OEIS
1370:OEIS
1350:/(10
1338:/(10
1328:OEIS
1133:1/Ɛ
933:1/9
896:OEIS
468:− 1)
387:/(10
164:7 =
152:7948
146:7948
110:= 7
3411:Ban
2799:or
2318:Lah
1515:doi
1511:105
1285:55
1265:14
1248:45
1217:35
1203:2Ɛ
1189:25
1175:1Ɛ
1090:In
1045:44
1031:13
1017:22
1003:58
958:28
944:18
464:(10
142:948
136:948
90:An
80:not
43:by
27:An
3481::
1529:,
1523:MR
1521:,
1509:,
1476:,
1453:,
1432:,
1411:,
1269:12
1242:9
1231:6
1225:8
1211:7
1197:6
1183:5
1169:4
1158:4
1152:3
1144:Ɛ
1138:2
1130:1
1127:1
1124:1
1071:≥
1039:9
1025:8
1011:7
997:6
986:6
980:5
972:6
966:4
952:3
938:2
930:1
927:1
924:1
898:)
557:0.
549:19
505:0.
497:19
470:.
424:10
405:10
375:≥
342:39
319:10
266:10
252:7.
233:0.
204:0.
132:48
126:48
1683:a
1564:e
1557:t
1550:v
1517::
1501:.
1460:.
1439:.
1418:.
1372:)
1352:n
1348:n
1346:-
1340:n
1336:n
1330:)
1313:m
1309:n
1307:/
1305:m
1301:m
1277:Ɛ
1257:ᘔ
1235:ᘔ
1162:7
1111:n
1105:n
1096:n
1080:n
1073:n
1069:k
1065:n
1061:n
990:7
911:n
905:n
878:n
862:n
815:k
811:n
615:n
611:n
607:n
587:m
570:.
554:=
546:2
518:.
502:=
494:1
479:n
475:n
466:n
453:m
439:)
436:1
428:m
420:(
414:1
408:n
401:k
389:n
385:k
377:n
373:k
368:k
364:n
347:.
339:7
334:=
331:x
315:x
312:+
309:7
303:=
300:x
297:4
271:.
246:=
230:=
227:x
224:4
201:=
192:=
189:x
168:.
122:8
116:8
108:k
104:n
100:n
96:k
92:n
57:n
45:n
31:-
29:n
20:)
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