This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A321612 #28 Jul 27 2023 08:19:31
%S A321612 2,4,6,7,9,13,16,21,31
%N A321612 Numbers k such that all k - t are triangular numbers where t is a triangular number in range k/2 <= t < k.
%C A321612 The following is a quotation from Hage-Hassan in his paper (see Link below). "The (concept of) right and left symmetry is fundamental in physics. This incites us to ask whether this symmetry is in (the) primes. Find the numbers n with a + a' = n. a, a' are primes and {a} are all the primes with: n/2 <= a < n and n = 2,3, ..."
%C A321612 This sequence is analogous to A320447. Instead of the sequence of primes it uses the sequence of triangular numbers (A000217). It is conjectured that the sequence is finite and full.
%H A321612 Mehdi Hage-Hassan, <a href="https://hal.archives-ouvertes.fr/hal-00879586/document">An elementary introduction to Quantum mechanic</a>, hal-00879586 2013 pp 58.
%e A321612 a(9) = 31, because the triangular numbers in the range 16 <= p < 31 are {21}. Also the complementary set {10} has all its members triangular numbers. This is the 9th occurrence of such a number.
%t A321612 TriangularQ[n_] := Module[{m=0}, While[n>m(m+1)/2, m++]; If[n==m(m+1)/2, True, False]]; plst[n_] := Select[Range[Ceiling[n/2], n-1], TriangularQ]; lst={}; Do[If[plst[n]!={}&&AllTrue[n-plst[n], TriangularQ], AppendTo[lst, n]], {n, 1, 200}]; lst
%Y A321612 Cf. A000217, A320447.
%K A321612 nonn,more
%O A321612 1,1
%A A321612 _Frank M Jackson_, Dec 18 2018