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happy numbers
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 
2714210704644 2 
3496174460791461889 3 
4354416353106359794351538 4 
5282117280600283012287609285516 5 
6249239225882230062246935234909231827 6 
7202901202506202280202871203167202900202229 7 
8177026176154180027176052177390176952179767175486 8 
9175385136193132629192577125335215636128212199263113624 9 
10143071138935141855140143140153139046141665141157147466145363 10 
11129588129162130086128144127727129801130744129032128835128248127487

A pictorial representation of the table above
motab
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.