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A (finite or infinite) strictly increasing sequence with the initial term $1$ is called an addition chain if each term after the initial one can be written as the sum of two earlier (not necessarily distinct) terms. For example,

$$a_1=1,\ a_2=1+1=2,\ a_3=2+2=4, \\ a_4=4+2=6,\ a_5=4+4=8,\ a_6=8+6=14$$

is an addition chain for $14$. For the basic knowledge about addition chains, one may consult the wiki article on addition chains available from http://en.wikipedia.org/wiki/Addition_chain .

For $x > 0$ let $\pi(x)$ denote the number of primes not exceeding $x$. Let us consider the sequence $$s_n = \pi\left(\frac{n(n+1)}2+1\right)\ \ (n = 1,2,3,...) \tag{$*$}.$$ The first 20 terms of this sequence are

$$1,\, 2,\, 4,\, 5,\, 6,\, 8,\, 10,\, 12,\, 14,\, 16,\, 19,\, 22,\, 24,\, 27,\, 30,\,33,\, 36,\, 39,\, 43,\, 47.$$

Question. Is the sequence $(*)$ an addition chain?

I formulated this question in 2015 (cf. http://oeis.org/A262446), and verified that for each $n=4,5,\ldots,10^5$ we have $s_n=s_k+s_m$ for some $1\le k<m<n$.

Your comments (including further check via a computer) are welcome!

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