cp's OEIS Frontend

A301517 Numbers whose ratio (sum of nonsquarefree divisors)/(sum of squarefree divisors) is a positive integer.

%I A301517 #34 Feb 21 2025 05:22:34
%S A301517 8,24,27,32,40,54,56,88,96,104,120,125,128,135,136,152,160,168,184,
%T A301517 189,216,224,232,243,248,250,264,270,280,296,297,312,328,343,344,351,
%U A301517 352,375,376,378,384,408,416,424,440,456,459,472,480,486,488,512,513,520
%N A301517 Numbers whose ratio (sum of nonsquarefree divisors)/(sum of squarefree divisors) is a positive integer.
%C A301517 Or numbers m such that r = A162296(m) / A048250(m) is a positive integer.
%C A301517 Conjecture: if r = A162296(a(n)) / A048250(a(n)) is a perfect square, r belongs to A001248.
%C A301517 The corresponding sequence b(n) = {r} begins with {4, 4, 9, 20, 4, 9, 4, 4, 20, 4, 4, 25, 84, 9, 4, 4, 20, 4, 4, 9, 49, 20, 4, 90, 4, 25, ... }. A majority of numbers of b(n) are perfect squares.
%C A301517 The numbers 2^(2n+1) with k > 0 are in the sequence (A004171).
%C A301517 The numbers prime(n)^3 are in the sequence (A030078).
%C A301517 The numbers 8*prime(n) with n > 1 are in the sequence.
%C A301517 Note that "positive integer", in the definition, eliminates squarefree numbers (A005117) from this sequence. - _Michel Marcus_, Mar 24 2018
%C A301517 From _Robert Israel_, Mar 29 2018: (Start)
%C A301517 If n is in the sequence, then so is n*p for any prime p coprime to n.
%C A301517 If m and n are in the sequence and are coprime, then m*n is in the sequence. (End)
%C A301517 The exponentially odd numbers (A268335) that are not squarefree are in the sequence. - _Amiram Eldar_, Jul 04 2020
%C A301517 The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 9, 99, 972, 9672, 96630, 966119, 9660732, 96606486, 966062725, ... . Apparently the asymptotic density of this sequence is 0.096606... . Note that most of the terms are in its subsequence A374459 whose asymptotic density is A065463 - A059956 = 0.096515099145... . - _Amiram Eldar_, Feb 20 2025
%H A301517 Robert Israel, <a href="/A301517/b301517.txt">Table of n, a(n) for n = 1..10000</a>
%e A301517 27 is in the sequence because A162296(27) / A048250(27) = 36/4 = 9.
%p A301517 filter:= proc(n) local S,N; uses numtheory;
%p A301517   S, N:= selectremove(issqrfree, divisors(n));
%p A301517   N <> {} and type(convert(N,`+`)/convert(S,`+`),integer)
%p A301517 end proc:
%p A301517 select(filter, [$1..1000]); # _Robert Israel_, Mar 29 2018
%t A301517 lst={};Do[If[DivisorSigma[1,n]-Total[Select[Divisors[n],SquareFreeQ]]>0&&IntegerQ[(DivisorSigma[1,n]-Total[Select[Divisors[n],SquareFreeQ]])/Total[Select[Divisors[n],SquareFreeQ]]],AppendTo[lst,n]],{n,520}];lst
%t A301517 rpiQ[n_]:=Module[{d=Divisors[n],sf,ot,ra},sf=Select[d,SquareFreeQ];ot=Complement[ d, sf];ra= Total[ ot]/Total[sf];ra>0&&IntegerQ[ra]]; Select[Range[600],rpiQ] (* _Harvey P. Dale_, Mar 19 2019 *)
%t A301517 f[p_, e_] := (p^(e + 1) - 1)/(p^2 - 1); ratio[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[2, 520], (r = ratio[#]) > 1 && IntegerQ[r] &] (* _Amiram Eldar_, Jul 04 2020 *)
%o A301517 (PARI) isok(n) = my(s = sumdiv(n, d, !issquarefree(d)*d)); s && !(s % (sigma(n) - s)); \\ _Michel Marcus_, Mar 24 2018
%Y A301517 Cf. A004171, A005117, A030078, A048250, A162296, A268335, A335989, A374459.
%Y A301517 Contains A056824.
%Y A301517 Cf. A059956, A065463.
%K A301517 nonn
%O A301517 1,1
%A A301517 _Michel Lagneau_, Mar 23 2018