Editing Happy number (section)

==Specific ''b''-happy numbers==
===4-happy numbers===
For <math>b = 4</math>, the only positive perfect digital invariant for <math>F_{2, b}</math> is the trivial perfect digital invariant 1, and there are no other cycles. Because all numbers are [[Periodic point#Iterated functions|preperiodic points]] for <math>F_{2, b}</math>, all numbers lead to 1 and are happy. As a result, [[base 4]] is a happy base.

===6-happy numbers===
For <math>b = 6</math>, the only positive perfect digital invariant for <math>F_{2, b}</math> 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 <math>F_{2, b}</math>, 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 = 6<sup>4</sup> 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, 1295

===10-happy numbers===
For <math>b = 10</math>, the only positive perfect digital invariant for <math>F_{2, b}</math> 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 <math>F_{2, b}</math>, 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 {{OEIS|id=A007770}}.

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. {{OEIS|id=A124095}}.

The first pair of consecutive 10-happy numbers is 31 and 32.<ref>{{cite OEIS |1=A035502 |2=Lower of pair of consecutive happy numbers |access-date=8 April 2011}}</ref> The first set of three consecutive is 1880, 1881, and 1882.<ref>{{cite OEIS |1=A072494 |2=First of triples of consecutive happy numbers |access-date=8 April 2011}}</ref> It has been proven that there exist sequences of consecutive happy numbers of any natural number length.<ref>{{Cite arXiv |title=Consecutive Happy Numbers |eprint=math/0607213 |last1=Pan |first1=Hao |year=2006}}</ref> The beginning of the first run of at least ''n'' consecutive 10-happy numbers for ''n''&nbsp;=&thinsp;1, 2, 3, ... is<ref name="Sloane-A055629">{{Cite OEIS |1=A055629 |2=Beginning of first run of at least ''n'' consecutive happy numbers}}</ref>
: 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."<ref>{{cite journal |last=Styer |first=Robert |year=2010 |page=5 |title=Smallest Examples of Strings of Consecutive Happy Numbers |journal=[[Journal of Integer Sequences]] |volume=13 |id=10.6.3 |url=https://cs.uwaterloo.ca/journals/JIS/VOL13/Styer/styer5.html |via=[[University of Waterloo]]}} Cited in {{harvtxt|Sloane "A055629"}}.</ref>

The number of 10-happy numbers up to 10<sup>''n''</sup> for 1&nbsp;≤&nbsp;''n''&nbsp;≤&nbsp;20 is<ref>{{Cite OEIS |1=A068571 |2=Number of happy numbers <= 10^n}}</ref>
: 3, 20, 143, 1442, 14377, 143071, 1418854, 14255667, 145674808, 1492609148, 15091199357, 149121303586, 1443278000870, 13770853279685, 130660965862333, 1245219117260664, 12024696404768025, 118226055080025491, 1183229962059381238, 12005034444292997294.