Weird Number Checker

Abundant but not semiperfect

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About Weird Number Checker

A weird number checker testing if n is abundant (σ(n) > 2n) but not semiperfect (no subset of proper divisors sums to n). Weird numbers are rare: 70, 836, 4030, 5830... Shows divisors, abundance, and subset sum analysis. Client-side.

Weird Number Checker Features

  • Weird check
  • Abundance
  • Subset sum
  • Divisors
  • Sequence
Weird number: abundant (sum of proper divisors > n) but no subset of proper divisors sums to n. The first weird number is 70: divisors 1,2,5,7,10,14,35 sum to 74>70, but no subset sums to exactly 70. Extremely rare: only 7 weird numbers below 10,000.

How to Use

Enter n:

  • Abundant?: Is σ(n)>2n?
  • Semiperfect?: Does subset sum to n?
  • Weird?: Abundant AND not semiperfect

Rarity

Weird numbers below 10000: 70, 836, 4030, 5830, 7192, 7912, 9272. Only 7! Below 10^6: about 24. All known weird numbers are even. Whether odd weird numbers exist is open.

Theory

  • Benkoski conjectured: no odd weird numbers
  • If n is weird, kn is weird for appropriate k
  • Primitive weird: not a multiple of smaller weird
  • Related to the coin problem and subset sum

Step-by-Step Instructions

  1. 1Enter number.
  2. 2Check abundance.
  3. 3Test subset sum.
  4. 4Determine weird.
  5. 5Browse known.

Weird Number Checker — Frequently Asked Questions

Why is 70 weird?+

Proper divisors: 1,2,5,7,10,14,35. Sum=74>70 (abundant ✓). Check all subsets: no combination sums to 70. For example: 35+14+10+7+5−1=70? No, that's adding not a valid subset. The closest are 35+14+10+7+5=71 and 35+14+10+7+2+1=69. None hit 70 exactly.

Do odd weird numbers exist?+

Unsolved! No odd weird number has ever been found. Benkoski conjectured none exist. If one exists, it must be > 10^17. The difficulty: odd abundant numbers are already rare, and most of those turn out to be semiperfect.

What makes subset sum hard here?+

Testing if any subset of divisors sums to n is a variant of the subset sum problem (NP-complete in general). For weird number detection, we use dynamic programming. The DP is feasible because the number of divisors is small, but the concept connects to computational complexity.

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