Weird number

3 years ago by in Algebra, Algebra, Arithmetic, Arithmetic, Featured

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 few weird numbers are

70836, 4030, 5830, 7192, 7912, 9272, 10430, … (sequence A006037 in OEIS).

Properties 

It has been shown that an infinite number of weird numbers exist; in fact, the sequence of weird numbers has positive asymptotic density.

It is not known if any odd weird numbers exist; if any do, they must be greater than 232 ≈ 4×109.

Sidney Kravitz has shown that for k a positive integer, Q a prime exceeding 2k, and

R=\frac{2^kQ-(Q+1)}{(Q+1)-2^k};

also prime and greater than 2k, then

n=2^{k-1}QR

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

n=2^{56}\cdot(2^{61}-1)\cdot153722867280912929\ \approx\ 2\cdot10^{52}.

If n is weird, and p is a prime greater than the sum of divisors σ(n), then pn is also weird.

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