A341475 - OEIS

%I #22 Oct 03 2023 01:32:54

%S 12,18,24,30,36,40,48,54,56,66,80,84,90,96,112,126,132,156,176,198,

%T 200,208,220,270,280,304,352,364,380,448,550,570,594,690,736,882,910,

%U 918,928,945,992,1026,1040,1120,1216,1372,1376,1488,1638,1696,1722,1782

%N 2-near-perfect numbers.

%C A number n is k-near-perfect if n is the sum of all but k of the proper divisors of n.  Perfect numbers are 0-near-perfect and sequence A181595 lists the 1-near-perfect numbers.

%H Michael S. Branicky, <a href="/A341475/b341475.txt">Table of n, a(n) for n = 1..2000</a>

%H Vedant Aryan, Dev Madhavani, Savan Parikh, Ingrid Slattery, and Joshua Zelinsky, <a href="https://arxiv.org/abs/2310.01305">On 2-Near Perfect Numbers</a>, arXiv:2310.01305 [math.NT], 2023.

%H Hùng Việt Chu, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Chu/chu26.html">Divisibility of Divisor Functions of Even Perfect Numbers</a>, J. Int. Seq., Vol. 24 (2021), Article 21.3.4.

%H Paul Pollack and Vladimir Shevelev, <a href="https://doi.org/10.1016/j.jnt.2012.06.008">On perfect and near-perfect numbers</a>, J. Number Theory 132 (2012), pp. 3037-3046; also on <a href="http://arxiv.org/abs/1011.6160">arXiv</a>, arXiv:1011.6160 [math.NT], 2010-2012.

%e 48 is 2-near-perfect because its proper divisors are {1, 2, 3, 4, 6, 8, 12, 16, 24} and 48 = 1+2+3+4+6+8+24.

%o (Python)

%o from sympy import divisors

%o def ok(n):

%o   proper_divs = divisors(n)[:-1]

%o   s = sum(proper_divs)

%o   if s - 3 < n: return False

%o   if s - sum(proper_divs[-2:]) > n: return False

%o   for i, c1 in enumerate(proper_divs[:-1]):

%o     if s - c1 - proper_divs[i+1] < n: return False

%o     if s - c1 - n in proper_divs[i+1:]: return True

%o   return False

%o def aupto(limit): return [m for m in range(1, limit+1) if ok(m)]

%o print(aupto(1782)) # _Michael S. Branicky_, Feb 21 2021

%Y Cf. A000396, A181595.

%K nonn

%O 1,1

%A _Jeffrey Shallit_, Feb 13 2021