In mathematics, a strange number is a natural number that is redundant but not semi-perfect [1] . In other words, the sum of the proper divisors (divisors, including 1, but not including themselves) of the number is greater than the number itself, but by adding a subset of the divisors you cannot get the number itself.
The smallest odd number is 70. Its divisors are: 1, 2, 5, 7, 10, 14, and 35; their sum is 74, but adding up a subset of the divisors cannot get 70. The number 12, for example, is redundant, but not strange, because the divisors 12 are 1, 2, 3, 4 and 6, the sum of which is 16; but 2 + 4 + 6 = 12.
The first few strange numbers [2] are 70, 836, 4030, 5830, 7192, 7912, 9272, 10 430, ... It was shown that there are an infinite number of strange numbers, and that the sequence of strange numbers has a positive asymptotic density [3] .
It is not known whether odd odd numbers exist; if they exist, then should be greater than 2 32 ≈ 4⋅10 9 [4] . As part of the yoyo @ home voluntary distributed computing project, the Odd Weird Search [5] subproject is working to find a similar number in the range up to 10 28 .
Stanley Kravitz showed that if - a whole positive - simple, and
- - simple then
- Is a strange number [6] .
Using this formula, he was able to find a large strange number
- .
Notes
- ↑ Benkoski, Stan. E2308 (in Problems and Solutions) // The American Mathematical Monthly : journal. - Vol. 79 , no. 7 . - P. 774 .
- ↑ A006037 sequence in OEIS
- ↑ Benkoski, Stan; Paul Erdős. On Weird and Pseudoperfect Numbers (English) // Mathematics of Computation : journal. - 1974. - April ( vol. 28 , no. 126 ). - P. 617-623 .
- ↑ 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. "
- ↑ Odd Weird Search
- ↑ Kravitz, Stanley. A search for large weird numbers // Journal of Recreational Mathematics : journal. - Baywood Publishing, 1976. - Vol. 9 , no. 2 . - P. 82-85 .