OEIS A002097 gives numbers that are not the sum of 3 nonzero triangular numbers. They are just seven numbers: 1, 2, 4, 6, 11, 20, 29.
The site says that the seven numbers are all without references. Could you let me know the reference or proof?
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$\begingroup$ Just for fun : the sum of these seven numbers is equal to $ 73 $ , which can be written $ p_{t_{t_{t_{p_{1}}}}} $ . $\endgroup$– Sylvain JULIENCommented Jul 17, 2017 at 22:06
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1$\begingroup$ I don't see how one could prove that with current technology. If we multiply by $8$ and add $3$, then one must write $8n+3$ as a sum of three squares (which are all necessarily odd) and the number of representations of this grows like a class number. On the other hand, we want to throw away the number of representations where one of the squares equals $1$ -- this is roughly the number of ways of writing $8n+2$ as a sum of two squares, which could grow like a divisor function. So this problem looks to me of a comparable difficulty to the Euler idoneal number problem. $\endgroup$– LuciaCommented Jul 17, 2017 at 22:51
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$\begingroup$ Of course the previous comment does show that there are only finitely many exceptions, and one can also bound the number of exceptions. However it is not clear how one could determine a complete list unconditionally. $\endgroup$– LuciaCommented Jul 17, 2017 at 23:05
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$\begingroup$ @Lucia Yes, I agree that's not clear. So I guessed someone wrote a paper about it. $\endgroup$– P.-S. ParkCommented Jul 17, 2017 at 23:29
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$\begingroup$ See mathoverflow.net/questions/90914/sums-of-three-non-zero-squares for the question of sums of three nonzero squares. $\endgroup$– Gerry MyersonCommented Jul 18, 2017 at 12:07
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