A258026 Numbers k such that prime(k+2) - 2*prime(k+1) + prime(k) < 0.

4, 6, 9, 11, 12, 16, 18, 19, 21, 24, 25, 27, 30, 32, 34, 37, 40, 42, 44, 47, 48, 51, 53, 56, 58, 59, 62, 63, 66, 68, 72, 74, 77, 80, 82, 84, 87, 88, 91, 92, 94, 97, 99, 101, 103, 106, 108, 111, 112, 114, 115, 119, 121, 125, 127, 128, 130, 132, 133, 135, 137

Offset: 1

Author

Clark Kimberling, Jun 05 2015

Keywords

• nonn
• easy

Comments

Positions of strict descents in the sequence of differences between primes. Partial sums of A333215. - Gus Wiseman, Mar 24 2020

Examples

			The prime gaps split into the following maximal weakly increasing subsequences: (1,2,2,4), (2,4), (2,4,6), (2,6), (4), (2,4,6,6), (2,6), (4), (2,6), (4,6,8), (4), (2,4), (2,4,14), ... Then a(n) is the n-th partial sum of the lengths of these subsequences. - _Gus Wiseman_, Mar 24 2020
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000
• Wikipedia, Longest increasing subsequence

Crossrefs

Partition of the positive integers: A064113, A258025, A258026;

Corresponding partition of the primes: A063535, A063535, A147812.

Adjacent terms differing by 1 correspond to strong prime quartets A054804.

The version for the Kolakoski sequence is A156242.

First differences are A333215 (if the first term is 0).

The version for strict ascents is A258025.

The version for weak ascents is A333230.

The version for weak descents is A333231.

Prime gaps are A001223.

Positions of adjacent equal prime gaps are A064113.

Weakly increasing runs of compositions in standard order are A124766.

Strictly decreasing runs of compositions in standard order are A124769.

Cf. A000040, A000720, A001221, A036263, A054819, A084758, A124765, A124768, A333212, A333213, A333214, A333256.

Programs

Mathematica

u = Table[Sign[Prime[n+2] - 2 Prime[n+1] + Prime[n]], {n, 1, 200}];
Flatten[Position[u, 0]]   (* A064113 *)
Flatten[Position[u, 1]]   (* A258025 *)
Flatten[Position[u, -1]]  (* A258026 *)
Accumulate[Length/@Split[Differences[Array[Prime,100]],LessEqual]]//Most (* Gus Wiseman, Mar 24 2020 *)

Python

from itertools import count, islice
from sympy import prime, nextprime
def A258026_gen(startvalue=1): # generator of terms >= startvalue
    c = max(startvalue,1)
    p = prime(c)
    q = nextprime(p)
    r = nextprime(q)
    for k in count(c):
        if p+r<(q<<1):
            yield k
        p, q, r = q, r, nextprime(r)
A258026_list = list(islice(A258026_gen(),20)) # Chai Wah Wu, Feb 27 2024