A number such that the repeated sum of the digit squares leads to 1. more
The first 600 happy numbers :
1,
7,
10,
13,
19,
23,
28,
31,
32,
44,
49,
68,
70,
79,
82,
86,
91,
94,
97,
100,
103,
109,
129,
130,
133,
139,
167,
176,
188,
190,
192,
193,
203,
208,
219,
226,
230,
236,
239,
262,
263,
280,
291,
293,
301,
302,
310,
313,
319,
320,
326,
329,
331,
338,
356,
362,
365,
367,
368,
376,
379,
383,
386,
391,
392,
397,
404,
409,
440,
446,
464,
469,
478,
487,
490,
496,
536,
556,
563,
565,
566,
608,
617,
622,
623,
632,
635,
637,
638,
644,
649,
653,
655,
656,
665,
671,
673,
680,
683,
694,
700,
709,
716,
736,
739,
748,
761,
763,
784,
790,
793,
802,
806,
818,
820,
833,
836,
847,
860,
863,
874,
881,
888,
899,
901,
904,
907,
910,
912,
913,
921,
923,
931,
932,
937,
940,
946,
964,
970,
973,
989,
998,
1000,
1003,
1009,
1029,
1030,
1033,
1039,
1067,
1076,
1088,
1090,
1092,
1093,
1112,
1114,
1115,
1121,
1122,
1125,
1128,
1141,
1148,
1151,
1152,
1158,
1177,
1182,
1184,
1185,
1188,
1209,
1211,
1212,
1215,
1218,
1221,
1222,
1233,
1247,
1251,
1257,
1258,
1274,
1275,
1277,
1281,
1285,
1288,
1290,
1299,
1300,
1303,
1309,
1323,
1330,
1332,
1333,
1335,
1337,
1339,
1353,
1366,
1373,
1390,
1393,
1411,
1418,
1427,
1444,
1447,
1448,
1457,
1472,
1474,
1475,
1478,
1481,
1484,
1487,
1511,
1512,
1518,
1521,
1527,
1528,
1533,
1547,
1557,
1572,
1574,
1575,
1578,
1581,
1582,
1587,
1599,
1607,
1636,
1663,
1666,
1670,
1679,
1697,
1706,
1717,
1724,
1725,
1727,
1733,
1742,
1744,
1745,
1748,
1752,
1754,
1755,
1758,
1760,
1769,
1771,
1772,
1784,
1785,
1796,
1808,
1812,
1814,
1815,
1818,
1821,
1825,
1828,
1841,
1844,
1847,
1851,
1852,
1857,
1874,
1875,
1880,
1881,
1882,
1888,
1900,
1902,
1903,
1920,
1929,
1930,
1933,
1959,
1967,
1976,
1992,
1995,
2003,
2008,
2019,
2026,
2030,
2036,
2039,
2062,
2063,
2080,
2091,
2093,
2109,
2111,
2112,
2115,
2118,
2121,
2122,
2133,
2147,
2151,
2157,
2158,
2174,
2175,
2177,
2181,
2185,
2188,
2190,
2199,
2206,
2211,
2212,
2221,
2224,
2242,
2245,
2254,
2257,
2258,
2260,
2275,
2285,
2300,
2306,
2309,
2313,
2331,
2333,
2338,
2339,
2360,
2369,
2383,
2390,
2393,
2396,
2417,
2422,
2425,
2448,
2452,
2455,
2457,
2458,
2471,
2475,
2478,
2484,
2485,
2487,
2511,
2517,
2518,
2524,
2527,
2528,
2542,
2545,
2547,
2548,
2554,
2555,
2557,
2568,
2571,
2572,
2574,
2575,
2581,
2582,
2584,
2586,
2602,
2603,
2620,
2630,
2639,
2658,
2685,
2693,
2714,
2715,
2717,
2725,
2741,
2745,
2748,
2751,
2752,
2754,
2755,
2771,
2784,
2800,
2811,
2815,
2818,
2825,
2833,
2844,
2845,
2847,
2851,
2852,
2854,
2856,
2865,
2874,
2881,
2899,
2901,
2903,
2910,
2919,
2930,
2933,
2936,
2963,
2989,
2991,
2998,
3001,
3002,
3010,
3013,
3019,
3020,
3026,
3029,
3031,
3038,
3056,
3062,
3065,
3067,
3068,
3076,
3079,
3083,
3086,
3091,
3092,
3097,
3100,
3103,
3109,
3123,
3130,
3132,
3133,
3135,
3137,
3139,
3153,
3166,
3173,
3190,
3193,
3200,
3206,
3209,
3213,
3231,
3233,
3238,
3239,
3260,
3269,
3283,
3290,
3293,
3296,
3301,
3308,
3310,
3312,
3313,
3315,
3317,
3319,
3321,
3323,
3328,
3329,
3331,
3332,
3338,
3346,
3351,
3355,
3356,
3364,
3365,
3367,
3371,
3376,
3380,
3382,
3383,
3391,
3392,
3436,
3456,
3463,
3465,
3466,
3506,
3513,
3531,
3535,
3536,
3546,
3553,
3560,
3563,
3564,
3602,
3605,
3607,
3608,
3616,
3620,
3629,
3634,
3635,
3637,
3643,
3645,
3646,
3650,
3653,
3654,
3661,
3664,
3667,
3670,
3673,
3676,
3680,
3689,
3692,
3698,
3706,
3709,
3713,
3731,
3736,
3760,
3763,
3766,
3779,
3789,
3790,
3797,
3798,
3803,
3806,
3823,
3830,
3832,
3833,
3860,
3869,
3879,
3896,
3897,
3901,
3902,
3907,
3910,
3913,
3920,
3923,
3926,
3931,
3932,
3962,
3968,
3970.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 1418854 values, from 1 to 10000000).
n\r | 0 | 1 |
2 | 714210 | 704644 | 2 |
3 | 496174 | 460791 | 461889 | 3 |
4 | 354416 | 353106 | 359794 | 351538 | 4 |
5 | 282117 | 280600 | 283012 | 287609 | 285516 | 5 |
6 | 249239 | 225882 | 230062 | 246935 | 234909 | 231827 | 6 |
7 | 202901 | 202506 | 202280 | 202871 | 203167 | 202900 | 202229 | 7 |
8 | 177026 | 176154 | 180027 | 176052 | 177390 | 176952 | 179767 | 175486 | 8 |
9 | 175385 | 136193 | 132629 | 192577 | 125335 | 215636 | 128212 | 199263 | 113624 | 9 |
10 | 143071 | 138935 | 141855 | 140143 | 140153 | 139046 | 141665 | 141157 | 147466 | 145363 | 10 |
11 | 129588 | 129162 | 130086 | 128144 | 127727 | 129801 | 130744 | 129032 | 128835 | 128248 | 127487 |
A pictorial representation of the table above
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.