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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 (list; graph; refs; listen; history; text; internal format)
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.)

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

Zhi-Wei 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

Zhi-Wei Sun, Sep 22 2015

STATUS

approved

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Last modified February 21 01:15 EST 2019. Contains 320364 sequences. (Running on oeis4.)