3 proper use of TeX

The exciting question on alternating sums of binomial coefficients triggered me to ask the following much simpler question (sorry if it is too simple, but I'm not a number theorist, so I must be overlooking something obvious).

Consider the sum: $$s_n := \sum_{i=1}^n (-1)^{i-1}\mbox{gcd}(n+1,i).$$-1)^{i-1}\gcd(n+1,i).$$Simple numerical experiments suggest the following: 1. If n=2p-1 for a prime p, then s_n = 1  as Kevin observed, this one follows immediately, because the gcd is either 1, 2, or p, and it is easy to see which term contributes which part (even, odd, etc.)) 2. If n is an odd number [edit: not of the form 2p-1, prime p] (see A166257), then s_n < 0 3. otherwise, of course, s_n=0. How should I prove this? Observe that the \alpha-generalization of s_n (i.e., summing \mbox{gcd}^\alpha terms instead) does not satisfy the above observations. 2 cleaned up the question The exciting question on alternating sums of binomial coefficients triggered me to ask the following much simpler question (sorry if it is too simple, but I'm not a number theorist, so I must be overlooking something obvious). Consider the sum:$$s_n := \sum_{i=1}^n (-1)^{i-1}\mbox{gcd}(n+1,i).$$Simple numerical experiments suggest the following: 1. If n=2p-1 for a prime p, then s_n = 1  as Kevin observed, this one follows immediately, because the gcd is either 1, 2, or p, and it is easy to see which term contributes which part (even, odd, etc.)) 2. If n is an odd number [edit: not of the form "prime(k)+phi(prime(k))" 2p-1, prime p] (see A166257), then s_n < 0 3. otherwise, of course, s_n=0. How should I prove this? Observe that the \alpha-generalization of s_n (i.e., summing \mbox{gcd}^\alpha terms instead) does not satisfy the above observations. 1 # Alternating sums of GCDs The exciting question on alternating sums of binomial coefficients triggered me to ask the following much simpler question (sorry if it is too simple, but I'm not a number theorist, so I must be overlooking something obvious). Consider the sum:$$s_n := \sum_{i=1}^n (-1)^{i-1}\mbox{gcd}(n+1,i).

Simple numerical experiments suggest the following:

1. If $n=2p-1$ for a prime $p$, then $s_n = 1$
2. If $n$ is an odd number not of the form "prime(k)+phi(prime(k))" (see A166257), then $s_n < 0$
3. otherwise, of course, $s_n=0$.

How should I prove this? Observe that the $\alpha$-generalization of $s_n$ (i.e., summing $\mbox{gcd}^\alpha$ terms instead) does not satisfy the above observations.