This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A318807 #19 Jan 10 2025 12:15:01
%S A318807 1,3,9,22,27,66,70,88,94,115,119,170,198,210,214,217,264,265,280,282,
%T A318807 310,322,345,357,376,382,385,497,510,517,527,594,630,642,651,679,680,
%U A318807 710,729,742,745,782,795,840,846,856,862,889,930,935,966,970,1035,1066
%N A318807 Numbers whose sum of squarefree divisors and sum of nonsquarefree divisors are both perfect squares.
%C A318807 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.
%C A318807 Or numbers m such that A048250(m) and A162296(m) are perfect squares.
%C A318807 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), ...
%C A318807 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).
%C A318807 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.
%H A318807 Robert Israel, <a href="/A318807/b318807.txt">Table of n, a(n) for n = 1..10000</a>
%e A318807 27 is in the sequence because A048250(27) = 4 and A162296(27) = 36 are both a perfect square.
%p A318807 filter:= proc(n) local F, SF, NSF, t;
%p A318807 F:= ifactors(n)[2];
%p A318807 SF:= mul(1+t[1],t=F);
%p A318807 if not issqr(SF) then return false fi;
%p A318807 NSF:= mul((1-t[1]^(1+t[2]))/(1-t[1]), t=F) - SF;
%p A318807 issqr(NSF);
%p A318807 end proc:
%p A318807 select(filter, [$1..2000]); # _Robert Israel_, Sep 05 2018
%t A318807 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
%t A318807 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 *)
%o A318807 (PARI) isok(n) = {my(sd=sumdiv(n, d, issquarefree(d)*d)); issquare(sd) && issquare(sigma(n) - sd);} \\ _Michel Marcus_, Sep 04 2018
%Y A318807 Cf. A000040, A001248, A013929, A048250, A162296.
%K A318807 nonn
%O A318807 1,2
%A A318807 _Michel Lagneau_, Sep 04 2018
%E A318807 Definition modified by _Harvey P. Dale_, Sep 13 2024