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Happy number
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=== Natural density of ''b''-happy numbers === By inspection of the first million or so 10-happy numbers, it appears that they have a [[natural density]] of around 0.15. Perhaps surprisingly, then, the 10-happy numbers do not have an asymptotic density. The upper density of the happy numbers is greater than 0.18577, and the lower density is less than 0.1138.<ref>{{Cite journal |title=On the Density of Happy Numbers |journal=Integers |volume=13 |issue=2 |arxiv=1110.3836 |last1=Gilmer |first1=Justin |year=2013 |page=2 |bibcode=2011arXiv1110.3836G}}</ref>
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