A392440 Numbers k > 1 with at least one nontrivial divisor 1 < d < k such that the largest proper divisor is the sum of a subset of the other nontrivial divisors of k.

12, 18, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 72, 78, 80, 84, 90, 96, 100, 102, 108, 112, 114, 120, 126, 132, 135, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 225, 228, 234, 240, 246, 252

Offset

1,1

Comments

Equivalently, let D_{p}(k) = {1 < d < L : d | k} where p is the smallest prime divisor of k and L = k/p, then k is a term if there exists a subset S of D_{p}(k) such that L = Sum(S).

We will refer to these numbers as 'p-composable'. Each p-composable number belongs to a unique class determined by its smallest prime factor p. For example, 135 is 3-composable, while 2275 is 5-composable. The smallest p-composable numbers appear in A392492. An integer k is a term of this sequence if and only if k is p-composable for some p. Composite numbers that are not p-composable are listed in A392500.

By definition, the index p is the smallest prime divisor of k. Consequently, all 2-composable numbers are even, while all p-composable numbers where p > 2 are odd. No prime number can be p-composable because if n is prime, then L = n/n = 1, making D_{p}(n) empty, and the empty sum cannot equal 1. Furthermore, if the predecessor of an odd prime is composable, it must be a 2-composable number.

The set of nonnegative integers is the union {0, 1} U A000040 U A392500 U this sequence.

Links

Table of n, a(n) for n=1..56.

Peter Luschny, Composable, rigid, and sparse numbers. A Python notebook.

Example

The first few p-classes of composable numbers start:
  p =  2 | 12, 18, 24, 30, 36, 40, 42, 48, ...
  p =  3 | 135, 225, 315, 345, 405, 495, ...
  p =  5 | 1645, 1925, 2275, 3185, 4025, ...
  p =  7 | 14651, 17017, 19019, 20111, ...
  p = 11 | 8041, 23881, 46189, 62491, ...

Prog

(Python)  # See also links.
# The function 'subset_sum_exists' is defined in A392652.
from sympy import divisors
def is_composable(n):
    if n < 4: return 0
    divs = divisors(n)[1:-1]
    if not divs: return 0
    if len(divs) == 2: return 0
    L = divs.pop()
    if sum(divs) < L: return 0
    return divs[0] if subset_sum_exists(divs, L) else 0
print([n for n in range(253) if is_composable(n)])

Crossrefs

Cf. A020639, A070824, A392438 (p=2), A392437 (p=3), A392492 (p minimal), A392439 (prime), A392500 (noncomposable).

Sequence in context: A297925 A341099 A175837 * A136446 A392438 A074726

Adjacent sequences: A392437 A392438 A392439 * A392441 A392442 A392443

Keyword

nonn

Author

Peter Luschny, Jan 14 2026

Status

approved