# repeating number, when n is multiple of 3.

If we take any integer multiple of $3$, and add up the cubes of its decimal digits, then take the result and sum the cubes of its digits, and so on, we invariably end up with $153.$

Why the above result is true for only an integer multiple of $3?$ can we prove the reasons for it? Is there any such numbers will invariably end with any number and not multiple of $3?$ If yes, how to find them.

A number is only divisible by three if the sum of its digits is a multiple of three. Since $a^3\equiv a\mod 3$, we see that after applying the 'step' (the function $s$) to a number divisible by three will still be divisible by three. Also, a number not divisible by $3$ will never be divisible by three after the step (because everything here works in two directions). Since $3|153$ (because $3|1+5+3=9$), only numbers divisible by $3$ can end in $153$.

In general, because $9^3<1000$, $s(n)<n$ when $n>10000$ and $s(n)<10000$ when $n\leq 10000$ (these are not a sharp bounds). This implies that the sequence $\left(s^k(n)\right)_{k\geq 0}$ will be decreasing until it falls below $10000$. Because of that, the function will eventually get in a finite loop. Thus, there is an $l$ such that, for a sufficient large $K$, $s^{k+l}(n)=s^k(n)$ for all $k>K$. It is possible to calculate all fixpoints $n$ of $s$ for which $s(n)=n$. I think this only applies to $153$, since it is given in this problem, but maybe there are some other numbers with this property.

EDIT As user44197 points out, the only other fixpoints (I assume) are $370$, $371$ and $407$. And of course $0$ and $1$, but they are trivial. As you can see, none of these numbers is divisible by $3$, so they can only end up in $153$ or in a cycle, but there aren't any cycles apparently. For other numbers then $3$, I don't know whether or not multiples will always end in the same number, but I think not. The only number that may be of interest is $11$, because a number is a multiple of $11$ when the alternating sum of its digits is a multiple of $11$. Problem is that $2^3=8\neq \pm 2\mod11$, so divisibility it not conserved. For any other interesting numbers, you can search for $n$ s.t. $i^3\equiv \pm i\mod n$ for all $1\leq i\leq n$. Thus, $2^3=8\equiv \pm 2\mod n$. This is true for $n=2$, $n=3$, $n=5$, $n=6$ and $n=10$. $n=2$ doesn't work, because $s(12)=1+8=9$ and $9$ is odd. Same for $n=10$: $s(10)=1+0=1$. For $n=6$, we get $s(12)=9$. For $n=5$, we get $s(10)=1$, so this won't work either. Thus, only multiples of $3$ have the property that they are still multiples of $3$ after applying $s$.

• What is not clear is why the divisibility by $3$? As you correctly point out, the divisibility or non-divisibility gets propagated. So if we start with a number not divisible by $3$ will we reach a fixed point? The possible fixed points are 370, 371 and 407. – user44197 Dec 30 '13 at 5:13
• @Ragnar! I could not understand that, why it is applicable only for the divisibility by 3? – zovi Dec 30 '13 at 5:26
• @user44197!Hello! It is still more quite interesting. How you came to know these numbers 370, 371,... Is there any method to find such numbers? – zovi Dec 30 '13 at 5:31
• yes, I have a method. It is called wikipedia :-) – user44197 Dec 30 '13 at 5:34
• @Ragnar!great and good observation. I loved it! – zovi Dec 31 '13 at 4:40

is not an answer but an observation which should inspire the Researchers to identify the patterns(if any)

First of all, some cycle converges to Armstrong Numbers (a special case of Narcissistic number).

Rest form very small cycles

$$1=[1, 10, 100, 112, 121, 211, 778, 787, 877, 1000, 1012, 1021, 1102, 1120, 1189, 1198, 1201, 1210, 1234, 1243, 1324, 1342, 1423, 1432, 1579, 1597, 1759, 1795, 1819, 1891, 1918, 1957, 1975, 1981, 2011, 2101, 2110, 2134, 2143, 2314, 2341, 2413, 2431, 2779, 2797, 2977, 3124, 3142, 3214, 3241, 3412, 3421, 4123, 4132, 4213, 4231, 4312, 4321]$$

$$250=[4, 13, 25, 28, 31, 40, 46, 52, 64, 82, 103, 130, 205, 208, 250, 256, 265, 280, 289, 298, 301, 310, 349, 394, 400, 406, 439, 448, 460, 484, 493, 502, 520, 526, 562, 589, 598, 604, 625, 640, 652, 802, 820, 829, 844, 859, 892, 895, 928, 934, 943, 958, 982, 985, 1003, 1030, 1111, 1144, 1222, 1246, 1249, 1264, 1294, 1300, 1333, 1348, 1366, 1384, 1414, 1426, 1429, 1438, 1441, 1456, 1462, 1465, 1483, 1492, 1546, 1564, 1624, 1636, 1642, 1645, 1654, 1663, 1777, 1834, 1843, 1924, 1942, 2005, 2008, 2050, 2056, 2065, 2080, 2089, 2098, 2122, 2146, 2149, 2164, 2194, 2212, 2221, 2245, 2254, 2257, 2266, 2275, 2416, 2419, 2425, 2452, 2461, 2479, 2491, 2497, 2500, 2506, 2524, 2527, 2542, 2560, 2572, 2605, 2614, 2626, 2641, 2650, 2662, 2689, 2698, 2725, 2749, 2752, 2794, 2800, 2809, 2869, 2890, 2896, 2908, 2914, 2941, 2947, 2968, 2974, 2980, 2986, 3001, 3010, 3049, 3094, 3100, 3133, 3148, 3166, 3184, 3313, 3331, 3409, 3418, 3445, 3454, 3481, 3490, 3544, 3556, 3565, 3616, 3655, 3661, 3667, 3676, 3766, 3814, 3841, 3904, 3940, 4000, 4006, 4039, 4048, 4060, 4084, 4093, 4114, 4126, 4129, 4138, 4141, 4156, 4162, 4165, 4183, 4192, 4216, 4219, 4225, 4252, 4261, 4279, 4291, 4297, 4309, 4318, 4345, 4354, 4381, 4390, 4408, 4411, 4435, 4444, 4453, 4459, 4480, 4495, 4516, 4522, 4534, 4543, 4549, 4555, 4561, 4594, 4600, 4612, 4615, 4621, 4651, 4729, 4792, 4804, 4813, 4831, 4840, 4903, 4912, 4921, 4927, 4930, 4945, 4954, 4972]$$

$$370=[7, 19, 34, 37, 43, 58, 67, 70, 73, 76, 85, 88, 91, 109, 118, 124, 139, 142, 145, 148, 154, 157, 166, 169, 175, 178, 181, 184, 187, 190, 193, 196, 214, 223, 226, 232, 241, 247, 259, 262, 268, 274, 277, 286, 295, 304, 307, 319, 322, 334, 340, 343, 346, 355, 358, 364, 367, 370, 376, 385, 391, 403, 412, 415, 418, 421, 427, 430, 433, 436, 451, 463, 466, 469, 472, 481, 496, 499, 508, 514, 517, 529, 535, 538, 541, 553, 556, 559, 565, 568, 571, 577, 580, 583, 586, 592, 595, 607, 616, 619, 622, 628, 634, 637, 643, 646, 649, 655, 658, 661, 664, 667, 670, 673, 676, 682, 685, 688, 691, 694, 700, 703, 706, 715, 718, 724, 727, 730, 736, 742, 751, 757, 760, 763, 766, 772, 775, 781, 799, 805, 808, 811, 814, 817, 826, 835, 841, 850, 853, 856, 862, 865, 868, 871, 880, 886, 889, 898, 901, 910, 913, 916, 925, 931, 946, 949, 952, 955, 961, 964, 979, 988, 994, 997, 1009, 1018, 1024, 1039, 1042, 1045, 1048, 1054, 1057, 1066, 1069, 1075, 1078, 1081, 1084, 1087, 1090, 1093, 1096, 1108, 1114, 1117, 1123, 1126, 1132, 1135, 1138, 1141, 1153, 1156, 1159, 1162, 1165, 1168, 1171, 1177, 1180, 1183, 1186, 1195, 1204, 1213, 1216, 1225, 1228, 1231, 1240, 1252, 1255, 1258, 1261, 1267, 1276, 1279, 1282, 1285, 1297, 1309, 1312, 1315, 1318, 1321, 1351, 1357, 1375, 1381, 1390, 1402, 1405, 1408, 1411, 1420, 1444, 1447, 1450, 1474, 1477, 1480, 1504, 1507, 1513, 1516, 1519, 1522, 1525, 1528, 1531, 1537, 1540, 1552, 1555, 1558, 1561, 1567, 1570, 1573, 1576, 1582, 1585, 1591, 1606, 1609, 1612, 1615, 1618, 1621, 1627, 1651, 1657, 1660, 1666, 1669, 1672, 1675, 1681, 1690, 1696, 1699, 1705, 1708, 1711, 1717, 1726, 1729, 1735, 1744, 1747, 1750, 1753, 1756, 1762, 1765, 1771, 1774, 1780, 1789, 1792, 1798, 1801, 1804, 1807, 1810, 1813, 1816, 1822, 1825, 1831, 1840, 1852, 1855, 1861, 1870, 1879, 1888, 1897, 1900, 1903, 1906, 1915, 1927, 1930, 1951, 1960, 1966, 1969, 1972, 1978, 1987, 1996, 2014, 2023, 2026, 2032, 2041, 2047, 2059, 2062, 2068, 2074, 2077, 2086, 2095, 2104, 2113, 2116, 2125, 2128, 2131, 2140, 2152, 2155, 2158, 2161, 2167, 2176, 2179, 2182, 2185, 2197, 2203, 2206, 2215, 2218, 2224, 2227, 2230, 2233, 2236, 2239, 2242, 2248, 2251, 2260, 2263, 2269, 2272, 2278, 2281, 2284, 2287, 2293, 2296, 2299, 2302, 2311, 2320, 2323, 2326, 2329, 2332, 2335, 2353, 2356, 2359, 2362, 2365, 2368, 2386, 2389, 2392, 2395, 2398, 2401, 2407, 2410, 2422, 2428, 2449, 2455, 2470, 2482, 2488, 2494, 2509, 2512, 2515, 2518, 2521, 2533, 2536, 2539, 2545, 2551, 2554, 2563, 2566, 2569, 2578, 2581, 2587, 2590, 2593, 2596, 2599, 2602, 2608, 2611, 2617, 2620, 2623, 2629, 2632, 2635, 2638, 2653, 2656, 2659, 2665, 2668, 2671, 2677, 2680, 2683, 2686, 2692, 2695, 2704, 2707, 2716, 2719, 2722, 2728, 2740, 2758, 2761, 2767, 2770, 2776, 2782, 2785, 2788, 2791, 2806, 2812, 2815, 2821, 2824, 2827, 2836, 2839, 2842, 2848, 2851, 2857, 2860, 2863, 2866, 2872, 2875, 2878, 2884, 2887, 2893, 2899, 2905, 2917, 2923, 2926, 2929, 2932, 2935, 2938, 2944, 2950, 2953, 2956, 2959, 2962, 2965, 2971, 2983, 2989, 2992, 2995, 2998, 3004, 3007, 3019, 3022, 3034, 3040, 3043, 3046, 3055, 3058, 3064, 3067, 3070, 3076, 3085, 3091, 3109, 3112, 3115, 3118, 3121, 3151, 3157, 3175, 3181, 3190, 3202, 3211, 3220, 3223, 3226, 3229, 3232, 3235, 3253, 3256, 3259, 3262, 3265, 3268, 3286, 3289, 3292, 3295, 3298, 3304, 3322, 3325, 3334, 3340, 3343, 3346, 3352, 3355, 3358, 3364, 3379, 3385, 3388, 3397, 3400, 3403, 3406, 3430, 3433, 3436, 3448, 3457, 3460, 3463, 3475, 3478, 3484, 3487, 3505, 3508, 3511, 3517, 3523, 3526, 3529, 3532, 3535, 3538, 3547, 3550, 3553, 3562, 3568, 3571, 3574, 3580, 3583, 3586, 3592, 3604, 3607, 3622, 3625, 3628, 3634, 3640, 3643, 3652, 3658, 3670, 3679, 3682, 3685, 3688, 3697, 3700, 3706, 3715, 3739, 3745, 3748, 3751, 3754, 3760, 3769, 3778, 3784, 3787, 3793, 3796, 3805, 3811, 3826, 3829, 3835, 3838, 3844, 3847, 3850, 3853, 3856, 3862, 3865, 3868, 3874, 3877, 3883, 3886, 3889, 3892, 3898, 3901, 3910, 3922, 3925, 3928, 3937, 3952, 3967, 3973, 3976, 3982, 3988, 4003, 4012, 4015, 4018, 4021, 4027, 4030, 4033, 4036, 4051, 4063, 4066, 4069, 4072, 4081, 4096, 4099, 4102, 4105, 4108, 4111, 4120, 4144, 4147, 4150, 4174, 4177, 4180, 4201, 4207, 4210, 4222, 4228, 4249, 4255, 4270, 4282, 4288, 4294, 4300, 4303, 4306, 4330, 4333, 4336, 4348, 4357, 4360, 4363, 4375, 4378, 4384, 4387, 4414, 4417, 4429, 4438, 4441, 4447, 4456, 4465, 4468, 4471, 4474, 4477, 4483, 4486, 4492, 4501, 4510, 4525, 4537, 4546, 4552, 4558, 4564, 4573, 4579, 4585, 4588, 4597, 4603, 4606, 4609, 4630, 4633, 4645, 4648, 4654, 4660, 4669, 4678, 4684, 4687, 4690, 4696, 4702, 4714, 4717, 4720, 4735, 4738, 4741, 4744, 4747, 4753, 4759, 4768, 4771, 4774, 4777, 4783, 4786, 4795, 4801, 4810, 4822, 4828, 4834, 4837, 4843, 4846, 4855, 4858, 4864, 4867, 4873, 4876, 4882, 4885, 4906, 4909, 4924, 4942, 4957, 4960, 4966, 4975, 4990, 4999]$$ $$160=[16, 22, 61, 106, 160, 202, 220, 601, 610, 1006, 1060, 1345, 1354, 1435, 1453, 1534, 1543, 1600, 2002, 2020, 2200, 2557, 2575, 2755, 3145, 3154, 3415, 3451, 3514, 3541, 3559, 3595, 3955, 4135, 4153, 4315, 4351, 4513, 4531, 4888]$$ $$407=[47, 74, 77, 89, 98, 407, 449, 470, 494, 578, 587, 668, 686, 704, 707, 740, 758, 770, 785, 788, 809, 857, 866, 875, 878, 887, 890, 908, 944, 980, 1124, 1139, 1142, 1148, 1157, 1175, 1178, 1184, 1187, 1193, 1214, 1241, 1319, 1367, 1376, 1391, 1412, 1418, 1421, 1481, 1517, 1559, 1571, 1589, 1595, 1598, 1637, 1673, 1688, 1715, 1718, 1736, 1751, 1763, 1781, 1814, 1817, 1841, 1859, 1868, 1871, 1886, 1895, 1913, 1931, 1955, 1958, 1985, 2114, 2141, 2249, 2294, 2333, 2369, 2378, 2387, 2396, 2411, 2429, 2477, 2492, 2558, 2585, 2588, 2639, 2693, 2738, 2747, 2774, 2783, 2837, 2855, 2858, 2873, 2885, 2924, 2936, 2942, 2963, 3119, 3167, 3176, 3191, 3233, 3269, 3278, 3287, 3296, 3323, 3332, 3359, 3377, 3395, 3539, 3593, 3617, 3629, 3671, 3692, 3716, 3728, 3737, 3761, 3773, 3782, 3827, 3872, 3911, 3926, 3935, 3953, 3962, 4007, 4049, 4070, 4094, 4112, 4118, 4121, 4181, 4211, 4229, 4277, 4292, 4409, 4448, 4484, 4490, 4577, 4700, 4727, 4757, 4772, 4775, 4811, 4844, 4889, 4898, 4904, 4922, 4940, 4988]$$ $$55=[55, 505, 550]$$ $$217=[79, 97, 115, 127, 151, 172, 217, 271, 388, 511, 709, 712, 721, 790, 838, 883, 907, 970, 1015, 1027, 1051, 1072, 1105, 1150, 1207, 1270, 1336, 1363, 1378, 1387, 1501, 1510, 1588, 1633, 1678, 1687, 1702, 1720, 1738, 1768, 1783, 1786, 1837, 1858, 1867, 1873, 1876, 1885, 2017, 2071, 2107, 2170, 2377, 2446, 2458, 2464, 2485, 2548, 2584, 2644, 2701, 2710, 2737, 2773, 2845, 2854, 3088, 3136, 3163, 3178, 3187, 3277, 3316, 3361, 3577, 3613, 3631, 3718, 3727, 3757, 3772, 3775, 3781, 3808, 3817, 3871, 3880, 4246, 4258, 4264, 4285, 4426, 4462, 4528, 4582, 4624, 4642, 4666, 4699, 4825, 4852, 4969, 4996]$$ $$133=[133, 313, 331, 679, 697, 769, 796, 967, 976, 1033, 1288, 1303, 1330, 1828, 1882, 1999, 2188, 2818, 2881, 3013, 3031, 3103, 3130, 3301, 3310, 3799, 3979, 3997]$$ $$352=[229, 235, 238, 253, 283, 292, 325, 328, 352, 382, 445, 454, 457, 475, 523, 532, 544, 547, 574, 745, 754, 823, 832, 922, 2029, 2035, 2038, 2053, 2083, 2092, 2209, 2290, 2305, 2308, 2338, 2350, 2380, 2383, 2503, 2530, 2803, 2830, 2833, 2902, 2920, 3025, 3028, 3052, 3082, 3205, 3208, 3238, 3250, 3280, 3283, 3328, 3382, 3466, 3502, 3520, 3646, 3664, 3802, 3820, 3823, 3832, 4045, 4054, 4057, 4075, 4366, 4405, 4450, 4504, 4507, 4540, 4570, 4636, 4663, 4705, 4750]$$ $$244=[244, 424, 442, 2044, 2347, 2374, 2404, 2437, 2440, 2473, 2734, 2743, 3247, 3274, 3337, 3373, 3427, 3472, 3724, 3733, 3742, 4024, 4042, 4204, 4237, 4240, 4273, 4327, 4372, 4402, 4420, 4723, 4732]$$

$$371=[2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 50, 53, 56, 59, 62, 65, 68, 71, 80, 83, 86, 92, 95, 101, 104, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 146, 149, 152, 155, 158, 161, 164, 167, 170, 173, 176, 179, 182, 185, 188, 191, 194, 197, 200, 203, 206, 209, 212, 215, 218, 221, 224, 227, 230, 233, 236, 239, 242, 245, 248, 251, 254, 257, 260, 263, 266, 269, 272, 275, 278, 281, 284, 287, 290, 293, 296, 299, 302, 305, 308, 311, 314, 317, 320, 323, 326, 329, 332, 335, 338, 341, 344, 347, 350, 353, 356, 359, 362, 365, 368, 371, 374, 377, 380, 383, 386, 389, 392, 395, 398, 401, 404, 410, 413, 416, 419, 422, 425, 428, 431, 434, 437, 440, 443, 446, 452, 455, 458, 461, 464, 467, 473, 476, 479, 482, 485, 488, 491, 497, 500, 503, 506, 509, 512, 515, 518, 521, 524, 527, 530, 533, 536, 539, 542, 545, 548, 551, 554, 557, 560, 563, 566, 569, 572, 575, 581, 584, 590, 593, 596, 599, 602, 605, 608, 611, 614, 617, 620, 623, 626, 629, 632, 635, 638, 641, 644, 647, 650, 653, 656, 659, 662, 665, 671, 674, 677, 680, 683, 689, 692, 695, 698, 701, 710, 713, 716, 719, 722, 725, 728, 731, 734, 737, 743, 746, 749, 752, 755, 761, 764, 767, 773, 776, 779, 782, 791, 794, 797, 800, 803, 806, 812, 815, 818, 821, 824, 827, 830, 833, 836, 839, 842, 845, 848, 851, 854, 860, 863, 869, 872, 881, 884, 893, 896, 899, 902, 905, 911, 914, 917, 920, 923, 926, 929, 932, 935, 938, 941, 947, 950, 953, 956, 959, 962, 965, 968, 971, 974, 977, 983, 986, 989, 992, 995, 998, 1001, 1004, 1007, 1010, 1013, 1016, 1019, 1022, 1025, 1028, 1031, 1034, 1037, 1040, 1043, 1046, 1049, 1052, 1055, 1058, 1061, 1064, 1067, 1070, 1073, 1076, 1079, 1082, 1085, 1088, 1091, 1094, 1097, 1100, 1103, 1106, 1109, 1112, 1115, 1118, 1121, 1127, 1130, 1133, 1136, 1145, 1151, 1154, 1160, 1163, 1166, 1169, 1172, 1181, 1190, 1196, 1199, 1202, 1205, 1208, 1211, 1217, 1220, 1223, 1226, 1229, 1232, 1235, 1238, 1244, 1247, 1250, 1253, 1256, 1259, 1262, 1265, 1268, 1271, 1274, 1277, 1280, 1283, 1286, 1289, 1292, 1295, 1298, 1301, 1304, 1307, 1310, 1313, 1316, 1322, 1325, 1328, 1331, 1334, 1337, 1340, 1343, 1346, 1349, 1352, 1355, 1358, 1361, 1364, 1370, 1373, 1379, 1382, 1385, 1388, 1394, 1397, 1400, 1403, 1406, 1409, 1415, 1424, 1427, 1430, 1433, 1436, 1439, 1442, 1445, 1448, 1451, 1454, 1457, 1460, 1463, 1466, 1469, 1472, 1475, 1478, 1484, 1487, 1490, 1493, 1496, 1499, 1502, 1505, 1508, 1511, 1514, 1520, 1523, 1526, 1529, 1532, 1535, 1538, 1541, 1544, 1547, 1550, 1553, 1556, 1562, 1565, 1568, 1574, 1577, 1580, 1583, 1586, 1592, 1601, 1604, 1607, 1610, 1613, 1616, 1619, 1622, 1625, 1628, 1631, 1634, 1640, 1643, 1646, 1649, 1652, 1655, 1658, 1661, 1664, 1667, 1670, 1676, 1679, 1682, 1685, 1691, 1694, 1697, 1700, 1703, 1706, 1709, 1712, 1721, 1724, 1727, 1730, 1733, 1739, 1742, 1745, 1748, 1754, 1757, 1760, 1766, 1769, 1772, 1775, 1778, 1784, 1787, 1790, 1793, 1796, 1799, 1802, 1805, 1808, 1811, 1820, 1823, 1826, 1829, 1832, 1835, 1838, 1844, 1847, 1850, 1853, 1856, 1862, 1865, 1874, 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• I've excluded the known pattern of $153$ which is also an Armstrong number – lab bhattacharjee Dec 30 '13 at 6:11
• bhattacherjee! Thank a lot! can you answer my question, why divisibility 3 is applicable to my post? – zovi Dec 30 '13 at 6:37
• What are these 'cycles'? And could you put them in an expendable box, so we don't have to scroll so far? – Ragnar Dec 30 '13 at 12:01
• @Ragnar, cycle = period, when the terms start to repeat themselves. Sorry, I don't know how to use an expendable box – lab bhattacharjee Dec 30 '13 at 14:44
• @labbhattacharjee! Sir, how you generated these many numbers. can you share the link please... – zovi Dec 31 '13 at 4:40