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

4, 28, 300, 3033, 30389, 303947, 3039643, 30396338, 303963527, 3039635535, 30396355364, 303963551074, 3039635509269, 30396355092700, 303963550926732, 3039635509266675, 30396355092702331, 303963550927021020

Offset: 1

Author

Frank M Jackson, Jun 18 2016

Keywords

• nonn
• more

Comments

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).

Links

• Keith Conrad, The Congruent Number Problem, The Harvard College Mathematics Review, (2008).
• Eric Weisstein's World of Mathematics, Squarefree
• Shou-Wu Zhang, The Congruent Numbers and Heegner Points, Asian Pacific Mathematics Newsletter, Vol 3(2) (2013).

Crossrefs

Cf. A006991, A062695, A071172, A104141, A273929, A274264.

Programs

Mathematica

Table[Length@Select[Range[10^n], MemberQ[{1, 2, 3}, Mod[#, 8]]&&SquareFreeQ[#] &], {n, 1, 8}]

Extensions

a(10)-a(11) from Giovanni Resta, Jun 19 2016

a(12)-a(18) from Hiroaki Yamanouchi, Dec 25 2016