

A262439


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


4



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
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OFFSET

1,2


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.  ZhiWei Sun, Jan 29 2024


REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. (Cf. Section C6 on addition chains.)
ZhiWei 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 ChinaJapan Seminar (Fukuoka, Oct. 28  Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169187.


LINKS



EXAMPLE

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


MATHEMATICA

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


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



