A392499 Numbers k > 1 with the smallest prime divisor p, where the sum of divisors d in the range 1 < d < k/p is at least k/p, but no subset of these divisors sums to k/p.

20, 45, 70, 88, 104, 105, 165, 175, 189, 195, 255, 272, 285, 304, 350, 368, 385, 441, 455, 464, 567, 572, 595, 650, 665, 693, 715, 735, 805, 819, 825, 836, 875, 1001, 1015, 1071, 1085, 1184, 1225, 1275, 1287, 1295, 1309, 1312, 1323, 1376, 1435, 1449, 1463, 1504

Offset

1,1

Comments

Equivalently, set D_{p}(k) = {1 < d < L : d | k}, where L = k/p. A term of this sequence is a noncomposable number (A392500) k such that Sum(D_{p}(k)) >= L, but no subset of D_{p}(k) sums to L.

We call these numbers 'rigid numbers' because, despite having enough admissible divisors 1 < d < k/p, their arithmetic structure prevents any subset from summing to k's largest proper divisor.

A392500 = this sequence U A392498; thus, noncomposable numbers are either sparse or rigid.

No term in this sequence is a prime number or a composable number (A392440).

Links

Table of n, a(n) for n=1..50.

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

Example

The first few p-classes of rigid composites start:
  p =  2 | 20, 70, 88, 104, 272, 304, ...
  p =  3 | 45, 105, 165, 189, 195, 255, ...
  p =  5 | 175, 385, 455, 595, 665, 715, ...
  p =  7 | 1001, 1309, 1463, 1547, 1729, ...
  p = 11 | 1573, 2431, 2717, 3289, 3553, ...

Prog

(Python)  # See also links.
# The function 'subset_sum_exists' is defined in A392652.
from sympy import divisors
def is_rigid(n: int) -> int:
    if n < 4: return 0
    divs = divisors(n)[1:-1]
    if not divs: return 0
    L = divs.pop()
    if sum(divs) < L: return 0
    return 0 if subset_sum_exists(divs, L) else divs[0]
print([n for n in range(1505) if is_rigid(n)])

Crossrefs

Cf. A392496 (p=2), A392495 (p=3), A392498 (sparse), A392500 (noncomposable), A392440 (composable), A392497 (minimal term), A006036 (primitive pseudoperfect).

Sequence in context: A256870 A134619 A219716 * A044097 A044478 A228319

Adjacent sequences: A392496 A392497 A392498 * A392500 A392501 A392502

Keyword

nonn

Author

Peter Luschny, Jan 16 2026

Status

approved