Nine Primitive All-Triangular Heronian Triangles Under Side 50,000
OEIS contributors should submit this sequence; recreational-mathematics catalogs of named-sequence-sided Heronian triangles should add the triangular case as a finite, exhaustively enumerated entry.
Description
Let T_n = n(n+1)/2 denote the n-th triangular number. We ask for every triangle whose three sides are all in the set {T_1, T_2, T_3, ...} and whose area is an integer. Enumerating all triples (a, b, c) with a ≤ b ≤ c, a, b, c ∈ {T_1, ..., T_315}, strict triangle inequality c < a + b, and 16·A² = (a+b+c)(-a+b+c)(a-b+c)(a+b-c) a nonzero perfect square divisible by 16, we tested 1,161,753 triples with T_k ≤ 49,770 and found 90 Heronian triangles, of which exactly 9 are primitive.
Purpose
Enumeration + structural split. The full ledger contains 90 all-triangular-sided Heronian triangles with max side ≤ 49,770. It is dominated by scalings of classical small Heronian triples (k·(a,b,c) where k·a, k·b, k·c all happen to be triangular numbers): 81 of the 90 are non-primitive. The scaling multipliers for the 81 non-primitives have a sharp peak at k = 3 (20 occurrences), followed by k = 21 (7), k = 33 (5), k = 5 (4), k = 7 (3), k = 6 (3). This concentration is explained by a simple density argument: k · a = T_n requires 2ka = n(n+1), which has integer solutions n for a large fraction of a when k shares many small factors with n(n+1). For k = 3 in particular, 6a = n(n+1) has a solution n for every a such that 24a + 1 is a perfect square; thus scaling a classical primitive Heronian triangle by 3 produces triangular-sided triangles at high density. The 9 primitive triangular-sided Heronian triangles are structurally meaningful, because they cannot be explained by scaling. They are: (91, 253, 300) at (T_13, T_22, T_24) with area 10,626; (276, 325, 595) at (T_23, T_25, T_34) area 12,558; (5151, 5671, 6328) at (T_101, T_106, T_112) area 13,841,520; (5995, 7503, 9316) at (T_109, T_122, T_136) area 22,448,976; (1891, 9316, 11175) at (T_61, T_136, T_149) area 1,767,000; (1378, 10585, 11781) at (T_52, T_145, T_153) area 3,819,816; (7503, 20503, 24310) at (T_122, T_202, T_220) area 71,411,340; (16653, 27028, 37675) at (T_182, T_232, T_274) area 200,149,950; (3081, 41041, 42778) at (T_78, T_286, T_292) area 53,287,080. None of the 9 are isoceles; they are all scalene. The smallest, (91, 253, 300), has factorization 91 = 7·13, 253 = 11·23, 300 = 2²·3·5², involving six distinct small primes and no shared factor. The fact that this is a new OEIS-absent sequence despite triangular numbers and Heronian triangles both being classical objects reflects how narrow the intersection is: among the roughly 1.16 million candidate triples tested, only 9 primitive solutions appear. The sequence gives a concrete, testable benchmark for future structural work — can the primitive density be bounded, or is there a generating family behind them?
Triangular numbers are the sizes of dot-triangles: 1 dot, then 1+2 = 3, then 1+2+3 = 6, then 10, 15, 21, 28, 36, 45, 55, 66, ... And a 'Heronian triangle' is a triangle whose three sides are whole numbers and whose area is also a whole number (like the 3-4-5 right triangle with area 6). Simple question: is there a triangle whose three sides are all triangular numbers and whose area is also a whole number? Yes — several. The simplest one with no common factor among the three sides is (91, 253, 300) — which is the 13th, 22nd, and 24th triangular numbers. Its area comes out to exactly 10,626. I enumerated every such triangle with the longest side at most 49,770 (the 315th triangular number) and found 90 of them, but almost all are 'boring' in a specific sense: 81 of the 90 are just scalings of smaller classical Heronian triangles. For example, 11 × (5, 5, 6) = (55, 55, 66), and 55 = T_10, 66 = T_11 are both triangular by coincidence, so (55, 55, 66) is on the list — but it's really just an enlarged version of (5, 5, 6), which nobody would count as 'new.' After removing those scalings, exactly 9 primitive (no common factor among the three sides) triangles remain. That 9 is the interesting number here: it's not zero (so the problem isn't trivially impossible), and it's not hundreds (so it's sparse enough to be worth cataloguing). I checked OEIS and the list does not appear there — not as max-sides, not as areas, not as triangular-number indices. It's a new small integer sequence that sits in the overlap of two classical families (triangular numbers; Heronian triangles) that apparently nobody has intersected before.
Novelty
I searched on 2026-04-13 for 'Heronian triangle triangular number sides' and related terms, and queried OEIS directly for the subsequences (91, 253, 300, 595, 6328) — no results; (10626, 12558, 1767000, 3819816) — no results; (300, 595, 6328, 9316, 11175, 11781, 24310) — no results. The Heronian-triangle Wikipedia article does not mention the triangular-number subcase. OEIS A334177 concerns consecutive-prime sides, not triangular. The sequence of 9 primitive Heronian triangles with triangular-number sides appears to be previously unrecorded.
How it upholds the rules
- 1. Not already discovered
- Targeted OEIS queries on the max-side, area, and index-triple prefixes all returned 'No results'. The Heronian-triangle Wikipedia article, the Ron Knott Surrey Fibonacci site, and the Singapore Math Society Medley survey do not mention the triangular-sided subcase.
- 2. Not computer science
- Classical number theory and Euclidean geometry. Objects: integer triangles whose sides are triangular numbers and whose area is rational and integer. Computer used only as exhaustive verifier over a finite enumeration.
- 3. Not speculative
- Each triangle is an exact Heron-formula computation, re-runnable from discovery/heronian_triangular.py. The enumeration is complete over its stated bound (max side ≤ 49,770), and extending the bound is a purely computational matter.
Verification
(1) The script enumerates every (i, j, k) with 1 ≤ i ≤ j ≤ k ≤ 315 and T_k < T_i + T_j and tests Heron's formula, terminating the innermost loop at the triangle-inequality cut. (2) Total all-triangular Heronian triangles found: 90. Primitives (gcd = 1): 9. (3) Independent sanity check: the smallest primitive (91, 253, 300) can be verified by hand: s = 322, A² = 322·231·69·22 = 112,911,876 = 10626². (4) Scaling consistency check: every non-primitive reduces under gcd division to a primitive Heronian triangle in the standard catalog — for example (55, 55, 66) = 11·(5, 5, 6), (28, 91, 105) = 7·(4, 13, 15), confirming no spurious entries. (5) OEIS independence: all three natural index sequences (max sides, areas, triangular indices) return 'no results' on OEIS as of 2026-04-13.
Sequences
(91, 253, 300) · (276, 325, 595) · (5151, 5671, 6328) · (5995, 7503, 9316) · (1891, 9316, 11175) · (1378, 10585, 11781) · (7503, 20503, 24310) · (16653, 27028, 37675) · (3081, 41041, 42778)
(13, 22, 24) · (23, 25, 34) · (101, 106, 112) · (109, 122, 136) · (61, 136, 149) · (52, 145, 153) · (122, 202, 220) · (182, 232, 274) · (78, 286, 292)
10,626 · 12,558 · 13,841,520 · 22,448,976 · 1,767,000 · 3,819,816 · 71,411,340 · 200,149,950 · 53,287,080
k=3: 20 triangles · k=21: 7 · k=33: 5 · k=5: 4 · k=7, k=6: 3 each · 9 other multipliers
Next steps
- Push the search to max side ≤ 10^6 to see whether the primitive density stabilizes (asymptotic constant) or grows, and whether a generating family emerges.
- Classify the 9 primitives by the 2-adic and 3-adic valuations of their sides; the density concentration of non-primitives at k = 3 suggests 3-adic structure may organize the primitives as well.
- Replace triangular numbers with tetrahedral numbers T_n^(3) = n(n+1)(n+2)/6 — are there primitive Heronian triangles with all tetrahedral sides?
- Submit the 9-term primitive sequence to OEIS under three keywords (triangular, Heronian, primitive).
Artifacts
- Triangular-sided Heronian enumeration script: discovery/heronian_triangular.py