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

1,1

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

Cf. A000396, A181595.

Sequence in context: A175837 A136446 A074726 * A091013 A348715 A159886

Adjacent sequences: A341472 A341473 A341474 * A341476 A341477 A341478

Keyword

nonn

Author

Jeffrey Shallit, Feb 13 2021

Status

approved