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A301517
Numbers whose ratio (sum of nonsquarefree divisors)/(sum of squarefree divisors) is a positive integer.
4
8, 24, 27, 32, 40, 54, 56, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 216, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 375, 376, 378, 384, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 512, 513, 520
OFFSET
1,1
COMMENTS
Or numbers m such that r = A162296(m) / A048250(m) is a positive integer.
Conjecture: if r = A162296(a(n)) / A048250(a(n)) is a perfect square, r belongs to A001248.
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.
The numbers 2^(2n+1) with k > 0 are in the sequence (A004171).
The numbers prime(n)^3 are in the sequence (A030078).
The numbers 8*prime(n) with n > 1 are in the sequence.
Note that "positive integer", in the definition, eliminates squarefree numbers (A005117) from this sequence. - Michel Marcus, Mar 24 2018
From Robert Israel, Mar 29 2018: (Start)
If n is in the sequence, then so is n*p for any prime p coprime to n.
If m and n are in the sequence and are coprime, then m*n is in the sequence. (End)
The exponentially odd numbers (A268335) that are not squarefree are in the sequence. - Amiram Eldar, Jul 04 2020
LINKS
EXAMPLE
27 is in the sequence because A162296(27) / A048250(27) = 36/4 = 9.
MAPLE
filter:= proc(n) local S, N; uses numtheory;
S, N:= selectremove(issqrfree, divisors(n));
N <> {} and type(convert(N, `+`)/convert(S, `+`), integer)
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 29 2018
MATHEMATICA
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
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 *)
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 *)
PROG
(PARI) isok(n) = my(s = sumdiv(n, d, !issquarefree(d)*d)); s && !(s % (sigma(n) - s)); \\ Michel Marcus, Mar 24 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Mar 23 2018
STATUS
approved