This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A262439 #39 Jan 30 2024 20:30:34
%S A262439 1,2,4,5,6,8,10,12,14,16,19,22,24,27,30,33,36,39,43,47,50,54,59,62,66,
%T A262439 70,75,79,84,90,94,99,102,108,115,121,126,131,137,142,149,154,161,167,
%U A262439 174,180,189,193,200,205,217,220,226,235,242,251,259,267,274,282,290,297,306,313,324,329,338,348,358,367
%N A262439 Number of primes not exceeding 1+n*(n+1)/2.
%C A262439 Conjecture: (i) The sequence is strictly increasing, and also a(n)^(1/n) > a(n+1)^(1/(n+1)) for all n = 3,4,....
%C A262439 (ii) The sequence is an addition chain. In other words, for each n = 2,3,... we have a(n) = a(k) + a(m) for some 0 < k <= m < n.
%C A262439 (iii) All the numbers Sum_{i=j..k} 1/a(i) with 0 < min{2,k} <= j <= k have pairwise distinct fractional parts.
%C A262439 See also A262446 related to part (ii) of this conjecture.
%C A262439 Concerning part (ii) of the conjecture, Neill Clift verified in 2024 that for all 1 < n <= 2^24 = 16777216 we have a(n) = a(k) + a(m) for some 0 < k <= m < n. - _Zhi-Wei Sun_, Jan 29 2024
%D A262439 R. K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. (Cf. Section C6 on addition chains.)
%D A262439 Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
%H A262439 Zhi-Wei Sun, <a href="/A262439/b262439.txt">Table of n, a(n) for n = 1..10000</a>
%H A262439 Neill Clift, <a href="http://additionchains.com/primes.html">Prime Count and Addition Chains</a>, 2024.
%H A262439 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.
%e A262439 a(3) = 4 since there are exactly four primes (namely, 2, 3, 5, 7) not exceeding 1 + 3*4/2 = 7.
%t A262439 a[n_]:=PrimePi[1+n(n+1)/2]
%t A262439 Do[Print[n," ",a[n]],{n,1,70}]
%Y A262439 Cf. A000040, A000217, A000720, A262403, A262446.
%K A262439 nonn
%O A262439 1,2
%A A262439 _Zhi-Wei Sun_, Sep 22 2015