Is the following conjecture true ?
$$\left\{\sum\limits_{\begin{array}{c}k=1\\k\in\mathbb{Z}\end{array}}^nk \ |\  n\in\Bbb Z\right\}  \cap
\left\lbrace \sum\limits_{\begin{array}{c}k=1\\k\in\mathbb{Z}\setminus\mathbb{P}\end{array}}^nk \ |\  n\in\mathbb{Z}\setminus\mathbb{P}\right\rbrace \cap
\left\{\sum\limits_{\begin{array}{c}k=2\\k\in\Bbb P\end{array}}^nk\ |\ n\in\Bbb P\right\}= \{ 28 \}$$

(that is : [A000217][1] $\cap$ [A051349][2] $\cap$ [A007504][3] = { 28 } )

I didn't find any other number below $10^{14}$ with this property ([Haskell script here](https://github.com/mmai/conjecture28/blob/master/haskell%2FC28.hs)). 

I originally posted this question here: [math.stackexchange/questions/1357530](http://math.stackexchange.com/questions/1357530/intersection-between-the-sums-of-the-first-integers-primes-and-non-primes), with an interesting contribution by [joriki](http://math.stackexchange.com/users/6622/joriki) in favor of the conjecture: 

> Unless there's a systematic reason for these sequences to coincide or avoid each other (which I doubt), we can estimate the number of triple coincidences of these three sequences by integrating over the product of their densities. The first one has density $1/n$ at $a_n=n(n+1)/2$, so at $x$ it has density $\sim(2x)^{−1/2}$. The others both omit numbers, so at given $x$ their densities are lower than this.

> Thus we can get an upper bound for the "probability" of there being such numbers beyond some $x_0$ from this integral:

> $\int_{x_0}^\infty\left(2x\right)^{-3/2}\mathrm dx=(2x_0)^{-1/2}$.

> As you've searched up to $10^{14}$, the "probability" of finding a triple coincidence beyond that is less that one in ten million. Unless there's a systematic reason...


  [1]: http://oeis.org/A000217
  [2]: http://oeis.org/A051349
  [3]: http://oeis.org/A007504