Vincenzo Manto

Planetary Numbers

Created: 6/8/2026 | Author: Vincenzo Manto and James C. McMahon , May 29 2026

Numbers m with exactly two distinct prime factors q and p such that m = q^k * p^j and p > q^k with k, j >= 1.

Formal Definition

Numbers m with exactly two distinct prime factors q and p such that:

m = qk · pj  where  p > qk  and  k, j ≥ 1

Structural Analysis & Motivation

The designation "Planetary Numbers" stems from a structural analogy based on the gravitational hierarchy of astronomical systems. In this framework, the prime factorization acts as a tiny celestial system:

  • The Stellar Core (The Central Mass): The dominant prime factor p represents a "heavy" central body.
  • The Satellites (The Orbital Group): The smaller prime power qk acts as a cluster of lesser satellites orbiting the massive center.

The mathematical constraint p > qk requires that the central "star" must always strictly exceed the combined weight of its internal "satellites" power cluster, enforcing a strict structural hierarchy inside the number's multiplicative decomposition.

Density and Asymptotic Behavior

For a fixed exponent k, the density of numbers satisfying p > qk tends to decrease as n increases. This behavior occurs because the dominant prime p must grow exponentially with respect to the smaller satellite base power qk to keep the system stable within the bounds.

Analytical Examples

  • 6 is included: Factors are 21 and 31. Here, p=3, q=2, k=1. Since 3 > 21, the system is valid.
  • 92 is included: Factors are 22 and 231. Here, p=23, q=2, k=2. Since 23 > 22 (23 > 4), it is valid.
  • 90 is excluded: 90 = 2 · 32 · 5. It contains three distinct prime factors, violating the binary system rule.
  • 89 is excluded: 89 is a prime number, meaning it lacks a companion satellite system.
  • 80 is excluded: 80 = 24 · 5. Here, p=5 and qk=24=16. Since 5 < 16, the satellite power overpowers the central prime.

Cross-References

See also OEIS entries: A001358 (Semiprimes), A007774.

© 2026 Vincenzo Manto. Conceptualized in collaboration with James C. McMahon.

Sequence Chart

Graph of A396597

Data

6,10,14,15,18,20,21,22,26,28,33,34,35,38,39,44,46,50,51,52,54,55,57,58,62,65,68,69,74,75,76,77,82,85,86,87,88,91,92,93,94,95,98,99,100,104,106,111,115,116,117,118,119,122,123,124,129,133,134,136,141,142

Computational Implementations

PYTHON

from sympy import factorint
def ok(n):
    factors = factorint(n)
    if len(factors) != 2:
        return False
    p1, p2 = factors.keys()
    pow1 = p1**factors[p1]
    pow2 = p2**factors[p2]
    return p2 > pow1 or p1 > pow2
print([m for m in range(1, 201) if ok(m)])

Mathematica

Select[Range[200], Length[f = FactorInteger[#]] == 2 && f[[2, 1]] > f[[1, 1]]^f[[1, 2]] &]

Cross-References

See also OEIS entries: Cf. A001358, A007774.