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

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

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

Links

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

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

Cf. A004171, A005117, A030078, A048250, A162296, A268335, A335989, A374459.

Contains A056824.

Cf. A059956, A065463.

Sequence in context: A376142 A377844 A240111 * A374459 A381312 A195086

Adjacent sequences: A301514 A301515 A301516 * A301518 A301519 A301520

Keyword

nonn,changed

Author

Michel Lagneau, Mar 23 2018

Status

approved