A318807 Numbers whose sum of squarefree divisors and sum of nonsquarefree divisors are both perfect squares.

1, 3, 9, 22, 27, 66, 70, 88, 94, 115, 119, 170, 198, 210, 214, 217, 264, 265, 280, 282, 310, 322, 345, 357, 376, 382, 385, 497, 510, 517, 527, 594, 630, 642, 651, 679, 680, 710, 729, 742, 745, 782, 795, 840, 846, 856, 862, 889, 930, 935, 966, 970, 1035, 1066

Offset: 1

Author

Michel Lagneau, Sep 04 2018

Keywords

• nonn

Comments

Let s be the sum of the squarefree divisors of a number m. The sequence lists the numbers m such that s and sigma(m) - s are both a perfect square.

Or numbers m such that A048250(m) and A162296(m) are perfect squares.

The corresponding pairs of squares (s, sigma(m) - s) are (1, 0), (4, 0), (4, 9), (36, 0), (4, 36), (144, 0), (144, 0), (36, 144), (144, 0), (144, 0), (144, 0), (324, 0), (144, 324), ...

The subsequence b(n) where s and sigma(m) - s are strictly positive begins with 9, 27, 88, 198, 264, 280, 376, 594, 630, ... b(n) is not squarefree (subsequence of A013929).

The subsequence c(n) where the ratio r = (sigma(a(n)) - s)/s is an integer begins with 27, 88, 264, 280, 376, 594, 680, 840, 856, 1128, 1240, ... and the corresponding r are 3^2, 2^2, 2^2, 2^2, 2^2, 3^2, 2^2, 2^2, 2^2, 2^2, 2^2, 2^2, 2^2, 5^2, 3^2, 2^2, 7^2, 3^2, 2^2, 11^2, ... It is conjectured that r belongs to A001248.

Examples

			27 is in the sequence because A048250(27) = 4 and A162296(27) = 36 are both a perfect square.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A001248, A013929, A048250, A162296.

Programs

Maple

filter:= proc(n) local F, SF, NSF, t;
  F:= ifactors(n)[2];
  SF:= mul(1+t[1],t=F);
  if not issqr(SF) then return false fi;
  NSF:= mul((1-t[1]^(1+t[2]))/(1-t[1]), t=F) - SF;
  issqr(NSF);
end proc:
select(filter, [$1..2000]); # Robert Israel, Sep 05 2018

Mathematica

lst={};Do[If[IntegerQ[Sqrt[Total[Select[Divisors[n],SquareFreeQ]]]]&&IntegerQ[Sqrt[DivisorSigma[1,n]-Total[Select[Divisors[n],SquareFreeQ]]]],AppendTo[lst,n]],{n,1100}];lst
sdsndQ[n_]:=Module[{d=Divisors[n],sf,nsf},sf=Select[d,SquareFreeQ];nsf= Complement[ d,sf];AllTrue[ {Sqrt[ Total[sf]],Sqrt[ Total[nsf]]},IntegerQ]]; Select[Range[1500],sdsndQ] (* Harvey P. Dale, Sep 13 2024 *)

PARI

isok(n) = {my(sd=sumdiv(n, d, issquarefree(d)*d)); issquare(sd) && issquare(sigma(n) - sd);} \\ Michel Marcus, Sep 04 2018

Extensions

Definition modified by Harvey P. Dale, Sep 13 2024