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Template:Sequence of the Day for September 10
Intended for: September 10, 2011
Timetable
- First draft entered by Alonso del Arte on May 31, 2011 (as a verbatim copy of a write-up from December 10, 2010) ✓
- Draft reviewed by Alonso del Arte on August 11, 2011 ✓
- Draft to be approved by August 10, 2011
| Yesterday's SOTD * Tomorrow's SOTD |
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- { 70, 836, 4030, 5830, 7192, 7912, 9272, ... }
It seems that the vast majority of abundant numbers (A005101) can be expressed as a sum of some of their proper divisors (they are pseudoperfect, listed in A005835), often in more than one way. For example, with 12, we have 12 = 1 + 2 + 3 + 6 (ignoring 4) or 12 = 2 + 4 + 6 (ignoring 1 and 3). Essentially, we figure out what the abundancy of the number is, figure out which divisors add up to that abundancy, and then add up the other divisors. Well, 70 is, like 12, an abundant number with proper divisors adding up to 4 more than itself. The eight divisors of 70 are {1, 2, 5, 7, 10, 14, 35, 70} (A018270). 1 and 2 add up to 3, one short of 4, and any sum of the other divisors is way too much. Isn’t that weird?
However, note that of the known weird numbers, none of them are odd. Robert Hearn has verified all weird numbers up to 10 17 are even. Also, a weird number can’t be S-perfect (a Granville number).
