

A321612


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


0




OFFSET

1,1


COMMENTS

The following is a quotation from HageHassan 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, ..."
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.


LINKS

Table of n, a(n) for n=1..9.
Mehdi HageHassan, An elementary introduction to Quantum mechanic, hal00879586 2013 pp 58.


EXAMPLE

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.


MATHEMATICA

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], n1], TriangularQ]; lst={}; Do[If[plst[n]!={}&&AllTrue[nplst[n], TriangularQ], AppendTo[lst, n]], {n, 1, 200}]; lst


CROSSREFS

Cf. A000217, A320447.
Sequence in context: A186708 A227697 A097457 * A259983 A050095 A102528
Adjacent sequences: A321609 A321610 A321611 * A321613 A321614 A321615


KEYWORD

nonn,more


AUTHOR

Frank M Jackson, Dec 18 2018


STATUS

approved



