1,1

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

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

27 is in the sequence because A162296(27) / A048250(27) = 36/4 = 9.

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

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 *)

(PARI) isok(n) = my(s = sumdiv(n, d, !issquarefree(d)*d)); s && !(s % (sigma(n) - s)); \\ Michel Marcus, Mar 24 2018

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

Contains A056824.

Cf. A059956, A065463.

Sequence in context: A376142 A377844 A240111 * A374459 A384520 A381312

Adjacent sequences: A301514 A301515 A301516 * A301518 A301519 A301520

Michel Lagneau, Mar 23 2018

approved