Happy numbers and perfect digital invariants
Formally, let be a natural number. Given the perfect digital invariant function :. for base , a number is -happy if there exists a such that , where represents the -th iteration of , and -unhappy otherwise. If a number is a nontrivial perfect digital invariant of , then it is -unhappy. For example, 19 is 10-happy, as : : : : For example, 347 is 6-happy, as : : : There are infinitely many -happy numbers, as 1 is a -happy number, and for every , ( in base ) is -happy, since its sum is 1. The ''happiness'' of a number is preserved by removing or inserting zeroes at will, since they do not contribute to the cross sum.Natural density of ''b''-happy numbers
By inspection of the first million or so 10-happy numbers, it appears that they have aHappy bases
A happy base is a number base where every number is -happy. The only happy bases less than are base 2 and base 4.Specific ''b''-happy numbers
4-happy numbers
For , the only positive perfect digital invariant for is the trivial perfect digital invariant 1, and there are no other cycles. Because all numbers are preperiodic points for , all numbers lead to 1 and are happy. As a result, base 4 is a happy base.6-happy numbers
For , the only positive perfect digital invariant for is the trivial perfect digital invariant 1, and the only cycle is the eight-number cycle : 5 → 41 → 25 → 45 → 105 → 42 → 32 → 21 → 5 → ... and because all numbers are preperiodic points for , all numbers either lead to 1 and are happy, or lead to the cycle and are unhappy. Because base 6 has no other perfect digital invariants except for 1, no positive integer other than 1 is the sum of the squares of its own digits. In base 10, the 74 6-happy numbers up to 1296 = 64 are (written in base 10): : 1, 6, 36, 44, 49, 79, 100, 160, 170, 216, 224, 229, 254, 264, 275, 285, 289, 294, 335, 347, 355, 357, 388, 405, 415, 417, 439, 460, 469, 474, 533, 538, 580, 593, 600, 608, 628, 638, 647, 695, 707, 715, 717, 767, 777, 787, 835, 837, 847, 880, 890, 928, 940, 953, 960, 968, 1010, 1018, 1020, 1033, 1058, 1125, 1135, 1137, 1168, 1178, 1187, 1195, 1197, 1207, 1238, 1277, 1292, 129510-happy numbers
For , the only positive perfect digital invariant for is the trivial perfect digital invariant 1, and the only cycle is the eight-number cycle : 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4 → ... and because all numbers are preperiodic points for , all numbers either lead to 1 and are happy, or lead to the cycle and are unhappy. Because base 10 has no other perfect digital invariants except for 1, no positive integer other than 1 is the sum of the squares of its own digits. In base 10, the 143 10-happy numbers up to 1000 are: : 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 . The distinct combinations of digits that form 10-happy numbers below 1000 are (the rest are just rearrangements and/or insertions of zero digits): : 1, 7, 13, 19, 23, 28, 44, 49, 68, 79, 129, 133, 139, 167, 188, 226, 236, 239, 338, 356, 367, 368, 379, 446, 469, 478, 556, 566, 888, 899. . The first pair of consecutive 10-happy numbers is 31 and 32. The first set of three consecutive is 1880, 1881, and 1882. It has been proven that there exist sequences of consecutive happy numbers of any natural number length. The beginning of the first run of at least ''n'' consecutive 10-happy numbers for ''n'' = 1, 2, 3, ... is : 1, 31, 1880, 7839, 44488, 7899999999999959999999996, 7899999999999959999999996, ... As Robert Styer puts it in his paper calculating this series: "Amazingly, the same value of N that begins the least sequence of six consecutive happy numbers also begins the least sequence of seven consecutive happy numbers." The number of 10-happy numbers up to 10''n'' for 1 ≤ ''n'' ≤ 20 is : 3, 20, 143, 1442, 14377, 143071, 1418854, 14255667, 145674808, 1492609148, 15091199357, 149121303586, 1443278000870, 13770853279685, 130660965862333, 1245219117260664, 12024696404768025, 118226055080025491, 1183229962059381238, 12005034444292997294.Happy primes
A -happy prime is a number that is both -happy and6-happy primes
In base 6, the 6-happy primes below 1296 = 64 are :211, 1021, 1335, 2011, 2425, 2555, 3351, 4225, 4441, 5255, 552510-happy primes
In base 10, the 10-happy primes below 500 are :7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193, 239, 263, 293, 313, 331, 367, 379, 383, 397, 409, 487 . The12-happy primes
In base 12, there are no 12-happy primes less than 10000, the first 12-happy primes are (the letters X and E represent the decimal numbers 10 and 11 respectively) :11031, 1233E, 13011, 1332E, 16377, 17367, 17637, 22E8E, 2331E, 233E1, 23955, 25935, 25X8E, 28X5E, 28XE5, 2X8E5, 2E82E, 2E8X5, 31011, 31101, 3123E, 3132E, 31677, 33E21, 35295, 35567, 35765, 35925, 36557, 37167, 37671, 39525, 4878E, 4X7X7, 53567, 55367, 55637, 56357, 57635, 58XX5, 5X82E, 5XX85, 606EE, 63575, 63771, 66E0E, 67317, 67371, 67535, 6E60E, 71367, 71637, 73167, 76137, 7XX47, 82XE5, 82EX5, 8487E, 848E7, 84E87, 8874E, 8X1X7, 8X25E, 8X2E5, 8X5X5, 8XX17, 8XX71, 8E2X5, 8E847, 92355, 93255, 93525, 95235, X1X87, X258E, X285E, X2E85, X85X5, X8X17, XX477, XX585, E228E, E606E, E822E, EX825, ...Programming example
The examples below implement the perfect digital invariant function for and a default base described in the definition of happy given at the top of this article, repeatedly; after each time, they check for both halt conditions: reaching 1, and repeating a number. A simple test in Python to check if a number is happy:See also
* Arithmetic dynamics * Fortunate number *References
Literature
*External links
* Schneider, Walter