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Advertisement: Try Blinkist for 7 days free and get 25% off the annual Premium subscription at https://www.blinkist.com/dorfuchs A good collection point for the current state of research is the English Wikipedia article: https://en.wikipedia.org/wiki/Happy_n... The proof that the upper density is greater than 0.18577 and the lower density is less than 0.1138 can be found here: Justin Gilmer - On the Density of Happy Numbers https://arxiv.org/abs/1110.3836 Special thanks to Silko Pillasch, who supports me on /dorfuchs. Patreon: / dorfuchs T-Shirts: http://www.DorFuchs.de/t-shirts/ Facebook: / dorfuchs Instagram: / dor.fuchs Twitter: / dorfuchs YouTube: / dorfuchs Website: http://www.DorFuchs.de/ Playlist with all math songs: http://bit.ly/MatheSongs Spotify: http://bit.ly/DorFuchsSpotify iTunes: http://bit.ly/DorFuchsiTunes This video was produced and published for private, non-commercial use and can be used free of charge for private purposes without consultation or written permission. Please note, however, that the video may not be changed in terms of content or graphics. When using it, please always state the DorFuchs YouTube channel as the source. Licensing is required for commercial use and for use for educational purposes requiring consent, such as public film screenings, public accessibility via educational servers, learning platforms or educational clouds, etc. Licenses are available from our sales partner http://www.filmsortiment.de. This video is suitable and intended for school teaching purposes and is therefore a protected work in accordance with §60a and §60b of the German Copyright Act.