A274264 Number of squarefree integers congruent to {5, 6, 7} mod 8 <= 10^n.

3, 33, 308, 3050, 30405, 303979, 3039648, 30396356, 303963597, 3039635407, 30396354916, 303963551200, 3039635509025, 30396355093247, 303963550927371, 3039635509273730, 30396355092701463, 303963550927001730

Offset: 1

Author

Frank M Jackson, Jun 16 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 {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).

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.

Programs

Mathematica

Table[Length@Select[Range[10^n], MemberQ[{5, 6, 7}, Mod[#, 8]]&& SquareFreeQ[#] &], {n, 1, 8}]

Extensions

a(10)-a(11) from Giovanni Resta, Jun 17 2016

a(7) corrected and a(12)-a(18) added by Hiroaki Yamanouchi, Dec 25 2016