OFFSET
1,1
COMMENTS
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.
LINKS
Michael S. Branicky, Table of n, a(n) for n = 1..2000
Vedant Aryan, Dev Madhavani, Savan Parikh, Ingrid Slattery, and Joshua Zelinsky, On 2-Near Perfect Numbers, arXiv:2310.01305 [math.NT], 2023.
Hùng Việt Chu, Divisibility of Divisor Functions of Even Perfect Numbers, J. Int. Seq., Vol. 24 (2021), Article 21.3.4.
Paul Pollack and Vladimir Shevelev, On perfect and near-perfect numbers, J. Number Theory 132 (2012), pp. 3037-3046; also on arXiv, arXiv:1011.6160 [math.NT], 2010-2012.
EXAMPLE
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.
PROG
(Python)
from sympy import divisors
def ok(n):
proper_divs = divisors(n)[:-1]
s = sum(proper_divs)
if s - 3 < n: return False
if s - sum(proper_divs[-2:]) > n: return False
for i, c1 in enumerate(proper_divs[:-1]):
if s - c1 - proper_divs[i+1] < n: return False
if s - c1 - n in proper_divs[i+1:]: return True
return False
def aupto(limit): return [m for m in range(1, limit+1) if ok(m)]
print(aupto(1782)) # Michael S. Branicky, Feb 21 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Jeffrey Shallit, Feb 13 2021
STATUS
approved