

A274264


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


3



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

1,1


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 SwinnertonDyer 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

Table of n, a(n) for n=1..18.
Keith Conrad, The Congruent Number Problem, The Harvard College Mathematics Review, (2008).
Eric Weisstein's World of Mathematics, Squarefree
ShouWu Zhang, The Congruent Numbers and Heegner Points, Asian Pacific Mathematics Newsletter, Vol 3(2) (2013).


MATHEMATICA

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


CROSSREFS

Cf. A006991, A062695, A071172, A104141, A273929.
Sequence in context: A190542 A180416 A043038 * A107127 A207323 A135697
Adjacent sequences: A274261 A274262 A274263 * A274265 A274266 A274267


KEYWORD

nonn,more


AUTHOR

Frank M Jackson, Jun 16 2016


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


STATUS

approved



