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 A274264 #16 Feb 16 2025 08:33:36
%S A274264 3,33,308,3050,30405,303979,3039648,30396356,303963597,3039635407,
%T A274264 30396354916,303963551200,3039635509025,30396355093247,
%U A274264 303963550927371,3039635509273730,30396355092701463,303963550927001730
%N A274264 Number of squarefree integers congruent to {5, 6, 7} mod 8 <= 10^n.
%C A274264 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.
%C A274264 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).
%H A274264 Keith Conrad, <a href="http://www.math.uconn.edu/~kconrad/articles/congruentnumber.pdf">The Congruent Number Problem</a>, The Harvard College Mathematics Review, (2008).
%H A274264 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree</a>
%H A274264 Shou-Wu Zhang, <a href="http://www.asiapacific-mathnews.com/03/0302/0012_0015.pdf">The Congruent Numbers and Heegner Points</a>, Asian Pacific Mathematics Newsletter, Vol 3(2) (2013).
%t A274264 Table[Length@Select[Range[10^n], MemberQ[{5, 6, 7}, Mod[#, 8]]&& SquareFreeQ[#] &], {n, 1, 8}]
%Y A274264 Cf. A006991, A062695, A071172, A104141, A273929.
%K A274264 nonn,more
%O A274264 1,1
%A A274264 _Frank M Jackson_, Jun 16 2016
%E A274264 a(10)-a(11) from _Giovanni Resta_, Jun 17 2016
%E A274264 a(7) corrected and a(12)-a(18) added by _Hiroaki Yamanouchi_, Dec 25 2016