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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 (list; graph; refs; listen; history; text; internal format)
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 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

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

Shou-Wu 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

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Last modified March 25 15:02 EDT 2019. Contains 321470 sequences. (Running on oeis4.)