TITLE: Intersection between the sums of the first positive integers, primes and non primes
QUESTION [7 upvotes]: Conjecture : 
$$\left\{\sum\limits_{\begin{array}{c}k=1\\k\in\mathbb{Z}\end{array}}^nk \ |\  n\in\Bbb Z\right\}  \cap
\left\lbrace \sum\limits_{\begin{array}{c}k=1\\k\in\mathbb{Z}\setminus\mathbb{P}\end{array}}^nk \ |\  n\in\mathbb{Z}\setminus\mathbb{P}\right\rbrace \cap
\left\{\sum\limits_{\begin{array}{c}k=2\\k\in\Bbb P\end{array}}^nk\ |\ n\in\Bbb P\right\}= \{ 28 \}$$
(that is : A000217 $\cap$ A051349 $\cap$ A007504 = { 28 } )
In other words : is 28 the only number that can be written as a sum of the first n positive integers (1+2+3+4+5+6+7), m first non primes (1+4+6+8+9) and p first primes (2+3+5+7+11) ?
I didn't find any other number below $10^{14}$ with this property (Haskell script here). 
EDIT : I cross-posted the question on https://mathoverflow.net/questions/212985/intersection-between-the-sums-of-the-first-integers-primes-and-non-primes

REPLY [10 votes]: 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...