A262439 Number of primes not exceeding 1+n*(n+1)/2.

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, 70, 75, 79, 84, 90, 94, 99, 102, 108, 115, 121, 126, 131, 137, 142, 149, 154, 161, 167, 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

Offset: 1

Author

Zhi-Wei Sun, Sep 22 2015

Keywords

• nonn

Comments

Conjecture: (i) The sequence is strictly increasing, and also a(n)^(1/n) > a(n+1)^(1/(n+1)) for all n = 3,4,....

(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.

(iii) All the numbers Sum_{i=j..k} 1/a(i) with 0 < min{2,k} <= j <= k have pairwise distinct fractional parts.

See also A262446 related to part (ii) of this conjecture.

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

Examples

			a(3) = 4 since there are exactly four primes (namely, 2, 3, 5, 7) not exceeding 1 + 3*4/2 = 7.
		

References

• R. K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. (Cf. Section C6 on addition chains.)
• 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.

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Neill Clift, Prime Count and Addition Chains, 2024.
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A000217, A000720, A262403, A262446.

Programs

Mathematica

a[n_]:=PrimePi[1+n(n+1)/2]
Do[Print[n," ",a[n]],{n,1,70}]