

A274043


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


1



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

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[{1, 2, 3}, Mod[#, 8]]&&SquareFreeQ[#] &], {n, 1, 8}]


CROSSREFS

Cf. A006991, A062695, A071172, A104141, A273929, A274264.
Sequence in context: A307083 A343709 A354147 * A007152 A345248 A295258
Adjacent sequences: A274040 A274041 A274042 * A274044 A274045 A274046


KEYWORD

nonn,more


AUTHOR

Frank M Jackson, Jun 18 2016


EXTENSIONS

a(10)a(11) from Giovanni Resta, Jun 19 2016
a(12)a(18) from Hiroaki Yamanouchi, Dec 25 2016


STATUS

approved



