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A274043 Number of squarefree integers congruent to {1, 2, 3} mod 8 <= 10^n.

%I A274043 #25 Feb 16 2025 08:33:36
%S A274043 4,28,300,3033,30389,303947,3039643,30396338,303963527,3039635535,
%T A274043 30396355364,303963551074,3039635509269,30396355092700,
%U A274043 303963550926732,3039635509266675,30396355092702331,303963550927021020
%N A274043 Number of squarefree integers congruent to {1, 2, 3} mod 8 <= 10^n.
%C A274043 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.
%C A274043 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).
%H A274043 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 A274043 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree</a>
%H A274043 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 A274043 Table[Length@Select[Range[10^n], MemberQ[{1, 2, 3}, Mod[#, 8]]&&SquareFreeQ[#] &], {n, 1, 8}]
%Y A274043 Cf. A006991, A062695, A071172, A104141, A273929, A274264.
%K A274043 nonn,more
%O A274043 1,1
%A A274043 _Frank M Jackson_, Jun 18 2016
%E A274043 a(10)-a(11) from _Giovanni Resta_, Jun 19 2016
%E A274043 a(12)-a(18) from _Hiroaki Yamanouchi_, Dec 25 2016