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Empirically, the limit of a(n)/10^n tends to 3/Pi^2 (A104141) and implies that the asymptotic density of squarefree numbers congruent to {5, 6, 7} mod 8 is half that of the asymptotic density of all squarefree integers (A071172). There is a slight bias towards more squarefree numbers congruent to {5, 6, 7} mod 8 that can be argued heuristically as {1, 2, 3} mod 8 contains a square residue and its equivalence class should contain less squarefree numbers.

Also it has been shown, conditional on the Birch Swinnerton-Dyer conjecture, that all squarefree integers congruent to {5, 6, 7} mod 8 (A273929) are primitive (squarefree) congruent numbers (A006991). However, this property applies only sparsely to squarefree integers congruent to {1, 2, 3} mod 8 (A062695).

Empirically, the limit of a(n)/10^n tends to 3/Pi^2 (A104141) and implies that the asymptotic density of squarefree numbers congruent to {1, 2, 3} mod 8 is half that of the asymptotic density of all squarefree integers (A071172). When this sequence is compared with squarefree numbers congruent to {5, 6, 7} mod 8 (A274264) it contains slightly fewer squarefree integers at each of the sampling points, 10^n for n > 1. It can be argued heuristically that, as {1, 2, 3} mod 8 contains a square residue, its equivalence class should contain fewer squarefree numbers.

Also it has been shown, conditional on the Birch Swinnerton-Dyer conjecture, that all squarefree integers congruent to {5, 6, 7} mod 8 (A273929) are primitive congruent numbers (A006991). However, this property applies only sparsely to squarefree integers congruent to {1, 2, 3} mod 8 (A062695).

Conjecture: the limit of a(n)/10^n tends to 3/Pi^2 (A104141). This is based on the assumption, conditional on the Birch Swinnerton-Dyer conjecture, that all squarefree integers congruent to {5, 6, 7} mod 8 (A273929) are a subset of primitive congruent numbers (A006991) and have a natural density of 3/Pi^2. However, squarefree integers congruent to {1, 2, 3} mod 8 are conjecturally sparsely congruent numbers with a natural density of 0. It has been proved without the BSD conjecture that the natural density of congruent numbers is at least 55.9% the natural density of squarefree numbers congruent to {5, 6, 7} mod 8 (see A. Smith link).

The Mathematica program below is a slow implementation of the Tunnell criteria for determining congruent numbers. It will give counts for up to 10^5 in realistic time. Counts for 10^6 and 10^7 have been derived from tables generated by Giovanni Resta (see link).

From Jose Aranda, Jul 04 2024: (Start)

The C++ program I have written calculates a(8) = 31925924 in 75 minutes. The time grows almost exponentially.

Looking at the 8 known terms I think the above conjecture should perhaps refer to A274264 rather than to the present sequence.

From the link "A trillion triangles": "The calculation found of these most mysterious congruent numbers up to a trillion = 3148379694."

That number corresponds to a(10) = 108744287 + A274264(10).

With A274264(10) = 3039635407. Now

3/Pi^2 = 0.303963550927013314...

A274264(08) = 0030396356.

A274264(10) = 003039635407.

A274264(18) = 00303963550927001730.

The sequence A274264 tends to this limit. This sequence may not. (End)

Conditional on the Birch and Swinnerton-Dyer conjecture, it can be shown that the only members of this sequence are the Fibonacci numbers {1,2,3,5}. The underlying pattern of three consecutive 1's per octet shows that numbers congruent to {5,6,7} mod 8 are congruent numbers. Also if n is a square then a(n)=5. This is because all congruent numbers can be obtained by multiplying a primitive congruent number A006991 by a positive square number A000290 and 5 is the least congruent number.