

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.


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

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.


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

Cf. A000040, A000217, A000720, A262403, A262446.
Sequence in context: A249025 A065502 A077255 * A091413 A020640 A068383
Adjacent sequences: A262436 A262437 A262438 * A262440 A262441 A262442


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Sep 22 2015


STATUS

approved



