A341475 2-near-perfect numbers.

12, 18, 24, 30, 36, 40, 48, 54, 56, 66, 80, 84, 90, 96, 112, 126, 132, 156, 176, 198, 200, 208, 220, 270, 280, 304, 352, 364, 380, 448, 550, 570, 594, 690, 736, 882, 910, 918, 928, 945, 992, 1026, 1040, 1120, 1216, 1372, 1376, 1488, 1638, 1696, 1722, 1782

Offset: 1

Author

Jeffrey Shallit, Feb 13 2021

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A000396, A181595.

Programs

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