|
| |
|
|
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.
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
|
|
|
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
|
|
|
|
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
|
| |
|
|
