What is...the fifteen theorem?

Goal.
I would like to tell you a bit about my favorite theorems, ideas or concepts in mathematics and why I like them so much.

This time.
What is...the fifteen theorem? Or: What is special about 1,2,3,5,6,7,10,14,15?

Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.

Slides.
http://www.dtubbenhauer.com/youtube.html

Conway on the 15 theorem.
https://www.youtube.com/watch?v=JybQ-6OTrLQ

Several square theorems.
https://brilliant.org/wiki/fermats-sum-of-two-squares-theorem/

The two square theorem.
https://en.wikipedia.org/wiki/Sum_of_two_squares_theorem
https://www.lukemastin.com/storyofmathematics/17th_fermat.html
https://people.mpim-bonn.mpg.de/zagier/files/doi/10.2307/2323918/fulltext.pdf
https://www.jstor.org/stable/2323918?origin=crossref
https://www.youtube.com/watch?v=DjI1NICfjOk

Numbers that are sums of two squares.
https://oeis.org/A001481

The three square theorem.
https://en.wikipedia.org/wiki/Legendre%27s_three-square_theorem
https://www.youtube.com/watch?v=m5evLoL0xwg

Numbers that are (not) sums of three squares.
https://oeis.org/A004215
https://oeis.org/A000408

The four square theorem.
https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem
https://www.britannica.com/science/Lagranges-four-square-theorem

Number of ways to write n a four squares.
https://oeis.org/A000118

The 15 and the 290 theorems.
https://www.theoremoftheday.org/NumberTheory/Fifteen/TotDFifteen.pdf
https://en.wikipedia.org/wiki/15_and_290_theorems
http://www.fen.bilkent.edu.tr/~franz/mat/15.pdf
http://www.wordpress.jonhanke.com/wp-content/uploads/2011/09/290-Theorem-preprint.pdf
https://mathworld.wolfram.com/FifteenTheorem.html

Mathematica.
https://demonstrations.wolfram.com/Fermats4n1TheoremAndTheNQueensProblem/
https://demonstrations.wolfram.com/RepresentationsByASumOfSquares/
https://demonstrations.wolfram.com/LagrangesFourSquareTheoremSeenUsingPolygonsAndLines/

Pictures used.
https://oeis.org/A001481
https://oeis.org/A004215
https://oeis.org/A000118
https://www.theoremoftheday.org/NumberTheory/Fifteen/TotDFifteen.pdf

is...thefifteentheorem?