Weird number

Key takeaways

Weird number belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Weird number to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Weird number from memory before moving on to harder problems.

Reference excerpt

In number theory, a weird number is a natural number that is abundant but not semiperfect. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself.

Examples The smallest weird number is 70. Its proper divisors are 1, 2, 5, 7, 10, 14, and 35; these sum to 74, but no subset of these sums to 70. The number 12, for example, is abundant but not weird, because the proper divisors of 12 are 1, 2, 3, 4, and 6, which sum to 16; but 2 + 4 + 6 = 12. The first several weird numbers are

70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, ... (sequence A006037 in the OEIS).

Properties

Infinitely many weird numbers exist. For example, 70p is weird for all primes p ≥ 149. In fact, the set of weird numbers has positive asymptotic density. It is not known if any odd weird numbers exist. If so, they must be greater than 1021. Sidney Kravitz has shown that for k a positive integer, Q a prime exceeding 2k, and

R = 2 k Q − ( Q + 1 ) ( Q + 1 ) − 2 k {\displaystyle R={\frac {2^{k}Q-(Q+1)}{(Q+1)-2^{k}}}}

also prime and greater than 2k, then

n = 2 k − 1 Q R {\displaystyle n=2^{k-1}QR}

is a weird number. With this formula, he found the large weird number

n = 2 56 ⋅ ( 2 61 − 1 ) ⋅ 153722867280912929 ≈ 2 ⋅ 10 52 . {\displaystyle n=2^{56}\cdot (2^{61}-1)\cdot 153722867280912929\ \approx \ 2\cdot 10^{52}.}

Primitive weird numbers A property of weird numbers is that if n is weird, and p is a prime greater than the sum of divisors σ(n), then pn is also weird. This leads to the definition of primitive weird numbers: weird numbers that are not a multiple of other weird numbers (sequence A002975 in the OEIS). Among the 1765 weird numbers less than one million, there are 24 primitive weird numbers. The construction of Kravitz yields primitive weird numbers, since all weird numbers of the form 2 k p q {\displaystyle 2^{k}pq} are primitive, but the existence of infinitely many k and Q which yield a prime R is not guaranteed. It is conjectured that there exist infinitely many primitive weird numbers, and Melfi has shown that the infinitude of primitive weird numbers is a consequence of Cramér's conjecture. Primitive weird numbers with as many as 16 prime factors and 14712 digits have been found.

Illustrations

Worked examples

Example 1 — a first encounter with Weird number

Start with the simplest possible case. Write down what Weird number claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Weird number before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Weird number ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Weird number

In research: Weird number appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Weird number in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Weird number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Divisor function, Integer sequences, so understanding it makes those chapters shorter.
In everyday life: Look for Weird number outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Weird number in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Weird number means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Weird number out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Weird number in simple terms?

In number theory, a weird number is a natural number that is abundant but not semiperfect. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself.

Why does Weird number matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Weird number?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Weird number.