Innumber theory, aweird number is anatural number that isabundant but notsemiperfect.^[1][2] In other words, the sum of theproper divisors (divisors including 1 but not itself) of the number is greater than the number, but nosubset of those divisors sums to the number itself.

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 butnot 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, ... (sequenceA006037 in theOEIS).

Infinitely many weird numbers exist.^[3] For example, 70p is weird for allprimesp ≥ 149. In fact, theset of weird numbers has positiveasymptotic density.^[4]

It is not known if anyodd weird numbers exist. If so, they must be greater than 10²¹.^[5]

Sidney Kravitz has shown that fork a positiveinteger,Q a prime exceeding 2^k, and

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

also prime and greater than 2^k, then

${\displaystyle n=2^{k-1}QR}$

is a weird number.^[6]With this formula, he found the large weird number

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

A property of weird numbers is that ifn is weird, andp is a prime greater than thesum of divisors σ(n), thenpn is also weird.^[4] This leads to the definition ofprimitive weird numbers: weird numbers that are not amultiple of other weird numbers (sequenceA002975 in theOEIS). 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${\displaystyle 2^{k}pq}$ are primitive, but the existence of infinitely manyk andQ which yield a primeR is not guaranteed. It isconjectured that there exist infinitely many primitive weird numbers, andMelfi has shown that the infinitude of primitive weird numbers is a consequence ofCramér's conjecture.^[7]Primitive weird numbers with as many as 16 prime factors and 14712 digits have been found.^[8]

Wikifunctions hasa weird number checking function.

• Untouchable number

• Weisstein, Eric W."Weird number".MathWorld.

• Divisor function
• Integer sequences

• Articles with short description
• Short description is different from Wikidata