Marc LeBrun wrote:
Recently I've been trying to exhort the Sequence Phanatiques to compute analogs of the sum-of-prime-factors (with and without multiplicity) in other arithmetics, such as Gaussian integers, GF(2), etc.
Don't worry, I will do the GF(2)[X] -thing, in my due time. (This is easy to do by using the existing the http://www.research.att.com/~njas/sequences/a091247.scm.txt Analogues for http://www.research.att.com/~njas/sequences/A000203 and similar sequences as well...)
I was wondering, specifically about GF(2), summing (ie XORing) the prime factors of N with multiplicity:
Noting that only the square-free part of N matters, since the square parts sum to 0...
A. Aside from the perfect squares (eg 5) are there any other N that sum to 0? Can they be characterized?
If we could just find from http://www.research.att.com/~njas/sequences/A014580 two successive terms, with first A014580(k) = 1 mod 4, and the second A014580(k+1) = A014580(k)+2, then by xoring them we could get 2, and the product A014580(k) x A014580(k+1) x 2 (where x would be GF(2)[X] multiplication: http://www.research.att.com/~njas/sequences/A048720 ) would satisfy the condition. However, skimming cursorily over the first 100 terms of http://www.research.att.com/~njas/sequences/A058943 I didn't find such a pair. (Maybe there's a reason for that? GF(2)[X] secrets elide me for a monent...) Yours, Antti