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A321612 Numbers k such that all k - t are triangular numbers where t is a triangular number in range k/2 <= t < k. 0

%I #28 Jul 27 2023 08:19:31

%S 2,4,6,7,9,13,16,21,31

%N Numbers k such that all k - t are triangular numbers where t is a triangular number in range k/2 <= t < k.

%C 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 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 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 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 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 Cf. A000217, A320447.

%K nonn,more

%O 1,1

%A _Frank M Jackson_, Dec 18 2018

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