A390186 Numbers k such that |s(k) - k| < |s(s(k)) - s(k)|, where s(k) is the sum of the proper divisors of k.

4, 8, 16, 18, 20, 24, 32, 36, 44, 45, 50, 52, 60, 63, 64, 68, 70, 72, 75, 78, 80, 88, 90, 96, 98, 100, 104, 105, 110, 114, 116, 120, 128, 130, 132, 136, 138, 144, 148, 152, 154, 160, 162, 165, 168, 170, 171, 184, 186, 189, 195, 196, 198, 200, 204, 208, 212, 216

Offset

1,1

Comments

The sequence is infinite, as it includes all numbers 2^m for m>1.

Primes and perfect numbers are not in the sequence.

Links

Table of n, a(n) for n=1..58.

Example

For k=4: s(4)=3, |3-4|=1; s(3)=1, |1-3|=2 >1, so included.
For k=6 (perfect): s(6)=6, |6-6|=0; s(6)=6, |6-6|=0, 0<0 false.
For k=7 (prime): s(7)=1, |1-7|=6; s(1)=0, |0-1|=1, 6<1 false.

Maple

s:= n-> numtheory[sigma](n)-n:
q:= k-> is(abs(s(k)-k)<abs(s(s(k))-s(k))):
select(q, [$1..250])[];  # Alois P. Heinz, Oct 31 2025

Mathematica

properDivSum[n_] := DivisorSigma[1, n] - n;
seq = {};
Do[
  sn = properDivSum[n];
  ssn = properDivSum[sn];
  If[Abs[sn - n] < Abs[ssn - sn],
    AppendTo[seq, n]
  ],
  {n, 2, 1000}
];
Print[seq];

Prog

(Python)
from sympy import divisor_sigma
def proper_div_sum(n): return divisor_sigma(n) - n
def isok(n):
    sn = proper_div_sum(n)
    ssn = proper_div_sum(sn)
    return abs(sn - n) < abs(ssn - sn)
print([n for n in range(2, 300) if isok(n)])
(PARI) s(n) = sigma(n) - n; \\ A001065
isok(k) = if (k>1, my(sk=s(k)); abs(sk - k) < abs(s(sk) - sk)); \\ Michel Marcus, Nov 02 2025

Crossrefs

Cf. A001065.

Sequence in context: A381076 A161994 A195065 * A312766 A312767 A312768

Adjacent sequences: A390183 A390184 A390185 * A390187 A390188 A390189

Keyword

nonn,easy

Author

Toby Walsh, Oct 28 2025

Status

approved