Weird number

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Divisibility-based
sets of integers
Form of factorization:
Prime number
Composite number
Powerful number
Square-free number
Achilles number
Constrained divisor sums:
Perfect number
Almost perfect number
Quasiperfect number
Multiply perfect number
Hyperperfect number
Superperfect number
Unitary perfect number
Semiperfect number
Primitive semiperfect number
Practical number
Numbers with many divisors:
Abundant number
Highly abundant number
Superabundant number
Colossally abundant number
Highly composite number
Superior highly composite number
Other:
Deficient number
Weird number
Amicable number
Friendly number
Sociable number
Solitary number
Sublime number
Harmonic divisor number
Frugal number
Equidigital number
Extravagant number
See also:
Divisor function
Divisor
Prime factor
Factorization
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In mathematics, a weird number is a natural number that is abundant but not semiperfect. [1] 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.

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 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, ... (sequence A006037 in OEIS). It has been shown that an infinite number of weird numbers exist, and the sequence of weird numbers has been proven to have positive asymptotic density.[2]

It is not known if any odd weird numbers exist; if any do, they must be greater than 232 ≈ 4×109.[3]

Stanley Kravitz has shown that if k is a positive integer, Q is a prime, and

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

is prime, then

n = 2k − 1QR

is a weird number. [4] With this formula, he was able to find the large weird number

n=2^{56}(2^{61}-1)153722867280912929\approx2\cdot10^{52}.

[edit] References

  1. ^ Benkoski, Stan (Aug.-Sep. 1972). "E2308 (in Problems and Solutions)". The American Mathematical Monthly 79 (7): 774. 
  2. ^ Benkoski, Stan; Paul Erdős (April 1974). "On Weird and Pseudoperfect Numbers". Mathematics of Computation 28 (126): 617-623. 
  3. ^ CN Friedman, "Sums of Divisors and Egyptian Fractions", Journal of Number Theory (1993). The result is attributed to "M. Mossinghoff at University of Texas - Austin".
  4. ^ Kravitz, Stanley (1976). "A search for large weird numbers". Journal of Recreational Mathematics 9 (2): 82-85. Baywood Publishing. 

[edit] External links