Helen Grundman has a written a number of articles about happy numbers. (I first heard of them at a talk of hers at a JMM.) References for her articles are listed below. I don't know if they discuss densities. One can also look at happy numbers to other bases, of course. According to Wikipedia: "The origin of happy numbers is not clear. Happy numbers were brought to the attention of Reg Allenby (a British author and Senior Lecturer in pure mathematics at Leeds University) by his daughter, who had learned of them at school. However, they 'may have originated in Russia'."

To answer Ricky Demer's question, yes, it's easily decidable, since if $N$ is large enough, then it's easy to see that $s(N)$ is a lot smaller than $N$. More precisely, $s(N)\le 81*\lceil\log_{10}(N)\rceil$. So one rapidly gets into and remains within a bounded range, after which one can check if the sequence cycles at 1, or in some other way.

• MR2382633 (2008m:11020) Grundman, H. G. ; Teeple, E. A. Sequences of consecutive happy numbers. Rocky Mountain J. Math. 37 (2007), no. 6, 1905--1916.
• MR2285991 (2007i:11016) Grundman, H. G. ; Teeple, E. A. Sequences of generalized happy numbers with small bases. J. Integer Seq. 10 (2007), no. 1, Article 07.1.8, 6 pp. (electronic).
• MR2022409 Grundman, H. G. ; Teeple, E. A. Heights of happy numbers and cubic happy numbers. Fibonacci Quart. 41 (2003), no. 4, 301--306.
• MR1866364 (2002h:11010) Grundman, H. G. ; Teeple, E. A. Generalized happy numbers. Fibonacci Quart. 39 (2001), no. 5, 462--466.

Added 18 October 2011: A relevant ArXiv post just appeared: On the Density of Happy Numbers, Justin Gilmer, http://arxiv.org/abs/1110.3836. The author proves that "happy numbers have upper density $\geq .18$ and lower density $\leq .12$."

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Helen Grundman has a written a number of articles about happy numbers. (I first heard of them at a talk of hers at a JMM.) References for her articles are listed below. I don't know if they discuss densities. One can also look at happy numbers to other bases, of course. According to Wikipedia: "The origin of happy numbers is not clear. Happy numbers were brought to the attention of Reg Allenby (a British author and Senior Lecturer in pure mathematics at Leeds University) by his daughter, who had learned of them at school. However, they 'may have originated in Russia'."

To answer Ricky Demer's question, yes, it's easily decidable, since if $N$ is large enough, then it's easy to see that $s(N)$ is a lot smaller than $N$. More precisely, $s(N)\le 81*\lceil\log_{10}(N)\rceil$. So one rapidly gets into and remains within a bounded range, after which one can check if the sequence cycles at 1, or in some other way.

• MR2382633 (2008m:11020) Grundman, H. G. ; Teeple, E. A. Sequences of consecutive happy numbers. Rocky Mountain J. Math. 37 (2007), no. 6, 1905--1916.
• MR2285991 (2007i:11016) Grundman, H. G. ; Teeple, E. A. Sequences of generalized happy numbers with small bases. J. Integer Seq. 10 (2007), no. 1, Article 07.1.8, 6 pp. (electronic).
• MR2022409 Grundman, H. G. ; Teeple, E. A. Heights of happy numbers and cubic happy numbers. Fibonacci Quart. 41 (2003), no. 4, 301--306.
• MR1866364 (2002h:11010) Grundman, H. G. ; Teeple, E. A. Generalized happy numbers. Fibonacci Quart. 39 (2001), no. 5, 462--466.