Zeta and Egyptian Fractions

What values of \( \xi (-n)\quad\) (where n = odd number i.e, 1, 3, 5, 7) ..... can not be represented by Egyptian fractions or Egyptian fractions - x (where x is any integer) form. I am only considering Egyptian fraction with length greater than 1.

  • For instance \( \xi (-1)\quad\) = -1/12 = [(1/2)+(1/3)+(1/12)] - 1
  • \( \xi (-17)\quad\) = 1/2+1/3+1/9+1/625+1/816137+1/1046695702500 - 4
  • \( \xi (-11)\quad\) = 691/32760 = (1/48)+(1/3855)+(1/16838640)

However if you see \( \xi (-3)\quad\) = 1/120 or \( \xi (-7)\quad\) = 1/240 they can not be represented as Egyptian faction of length > 1 Also note that since we are only considering Egyptian fraction with length greater than 1 \( \xi (-33)\quad\) which equals (1/12) -12635724796 also becomes part of this sequence

So If we start creating a sequence S(n) for which this is not possible then 3, 7 and 33 are first few candidates. I would like to learn more about this sequence. Also is there any way to know, for what values of n, \( \xi (-n)\quad\) will be positive?

Note by Vikram Pandya
3 years, 1 month ago

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