# Is the sequence $\left(\pi\left(\frac{n(n+1)}2+1\right)\right)_{n\ge1}$ an addition chain?

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.