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%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
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