A333214 -id:A333214 - cp's OEIS frontend

A238279 Triangle read by rows: T(n,k) is the number of compositions of n into nonzero parts with k parts directly followed by a different part, n>=0, 0<=k<=A004523(n-1).

1, 1, 2, 2, 2, 3, 4, 1, 2, 10, 4, 4, 12, 14, 2, 2, 22, 29, 10, 1, 4, 26, 56, 36, 6, 3, 34, 100, 86, 31, 2, 4, 44, 148, 200, 99, 16, 1, 2, 54, 230, 374, 278, 78, 8, 6, 58, 322, 680, 654, 274, 52, 2, 2, 74, 446, 1122, 1390, 814, 225, 22, 1, 4, 88, 573, 1796, 2714, 2058, 813, 136, 10, 4, 88, 778, 2694, 4927

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 22 2014

Same as A238130, with zeros omitted.
Last elements in rows are 1, 1, 2, 2, 1, 4, 2, 1, 6, 2, 1, 8, ... with g.f. -(x^6+x^4-2*x^2-x-1)/(x^6-2*x^3+1).
For n > 0, also the number of compositions of n with k + 1 runs. - Gus Wiseman, Apr 10 2020

			Triangle starts:
  00:  1;
  01:  1;
  02:  2;
  03:  2,   2;
  04:  3,   4,   1;
  05:  2,  10,   4;
  06:  4,  12,  14,    2;
  07:  2,  22,  29,   10,    1;
  08:  4,  26,  56,   36,    6;
  09:  3,  34, 100,   86,   31,    2;
  10:  4,  44, 148,  200,   99,   16,    1;
  11:  2,  54, 230,  374,  278,   78,    8;
  12:  6,  58, 322,  680,  654,  274,   52,    2;
  13:  2,  74, 446, 1122, 1390,  814,  225,   22,   1;
  14:  4,  88, 573, 1796, 2714, 2058,  813,  136,  10;
  15:  4,  88, 778, 2694, 4927, 4752, 2444,  618,  77,  2;
  16:  5, 110, 953, 3954, 8531, 9930, 6563, 2278, 415, 28, 1;
  ...
Row n=5 is 2, 10, 4 because in the 16 compositions of 5
  ##:  [composition]  no. of changes
  01:  [ 1 1 1 1 1 ]   0
  02:  [ 1 1 1 2 ]   1
  03:  [ 1 1 2 1 ]   2
  04:  [ 1 1 3 ]   1
  05:  [ 1 2 1 1 ]   2
  06:  [ 1 2 2 ]   1
  07:  [ 1 3 1 ]   2
  08:  [ 1 4 ]   1
  09:  [ 2 1 1 1 ]   1
  10:  [ 2 1 2 ]   2
  11:  [ 2 2 1 ]   1
  12:  [ 2 3 ]   1
  13:  [ 3 1 1 ]   1
  14:  [ 3 2 ]   1
  15:  [ 4 1 ]   1
  16:  [ 5 ]   0
there are 2 with no changes, 10 with one change, and 4 with two changes.

Joerg Arndt and Alois P. Heinz, Rows n = 0..180, flattened

Columns k=0-10 give: A000005 (for n>0), 2*A002133, A244714, A244715, A244716, A244717, A244718, A244719, A244720, A244721, A244722.
Row lengths are A004523.
Row sums are A011782.
The version counting adjacent equal parts is A106356.
The version for ascents/descents is A238343.
The version for weak ascents/descents is A333213.
The k-th composition in standard-order has A124762(k) adjacent equal parts, A124767(k) maximal runs, A333382(k) adjacent unequal parts, and A333381(k) maximal anti-runs.
Cf. A064113, A333214, A333216.

b:= proc(n, v) option remember; `if`(n=0, 1, expand(
      add(b(n-i, i)*`if`(v=0 or v=i, 1, x), i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..14);

b[n_, v_] := b[n, v] = If[n == 0, 1, Expand[Sum[b[n-i, i]*If[v == 0 || v == i, 1, x], {i, 1, n}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Maple *)
Table[If[n==0,1,Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[Split[#]]==k+1&]]],{n,0,12},{k,0,If[n==0,0,Floor[2*(n-1)/3]]}] (* Gus Wiseman, Apr 10 2020 *)

T_xy(max_row) = {my(N=max_row+1, x='x+O('x^N),h=(1+ sum(i=1,N,(x^i-y*x^i)/(1+y*x^i-x^i)))/(1-sum(i=1,N, y*x^i/(1+y*x^i-x^i)))); for(n=0,N-1, print(Vecrev(polcoeff(h,n))))}
T_xy(16) \\ John Tyler Rascoe, Jul 10 2024

G.f.: A(x,y) = ( 1 + Sum_{i>0} ((x^i)*(1 - y)/(1 + y*x^i - x^i)) )/( 1 - Sum_{i>0} ((y*x^i)/(1 + y*x^i - x^i)) ). - John Tyler Rascoe, Jul 10 2024

A106356 Triangle T(n,k) 0<=k

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 4, 3, 0, 1, 7, 6, 2, 0, 1, 14, 7, 8, 2, 0, 1, 23, 20, 10, 8, 2, 0, 1, 39, 42, 22, 13, 9, 2, 0, 1, 71, 72, 58, 28, 14, 10, 2, 0, 1, 124, 141, 112, 72, 33, 16, 11, 2, 0, 1, 214, 280, 219, 150, 92, 36, 18, 12, 2, 0, 1, 378, 516, 466, 311, 189, 112, 40, 20, 13, 2, 0, 1

Offset: 1

Christian G. Bower, Apr 29 2005

For n > 0, also the number of compositions of n with k + 1 maximal anti-runs (sequences without adjacent equal terms). - Gus Wiseman, Mar 23 2020

			T(4,1) = 3 because the compositions of 4 with 1 adjacent equal part are 1+1+2, 2+1+1, 2+2.
Triangle begins:
   1;
   1,  1;
   3,  0,  1;
   4,  3,  0, 1;
   7,  6,  2, 0, 1;
  14,  7,  8, 2, 0, 1;
  23, 20, 10, 8, 2, 0, 1;
  ...
From _Gus Wiseman_, Mar 23 2020 (Start)
Row n = 6 counts the following compositions (empty column shown by dot):
  (6)     (33)    (222)    (11112)  .  (111111)
  (15)    (114)   (1113)   (21111)
  (24)    (411)   (1122)
  (42)    (1131)  (2211)
  (51)    (1221)  (3111)
  (123)   (1311)  (11121)
  (132)   (2112)  (11211)
  (141)           (12111)
  (213)
  (231)
  (312)
  (321)
  (1212)
  (2121)
(End)

Alois P. Heinz, Rows n = 1..141, flattened
A. Knopfmacher and H. Prodinger, On Carlitz compositions, European Journal of Combinatorics, Vol. 19, 1998, pp. 579-589.

Row sums: 2^(n-1)=A000079(n-1). Columns 0-4: A003242, A106357-A106360.
The version counting adjacent unequal parts is A238279.
The k-th composition in standard-order has A124762(k) adjacent equal parts and A333382(k) adjacent unequal parts.
The k-th composition in standard-order has A124767(k) maximal runs and A333381(k) maximal anti-runs.
The version for ascents/descents is A238343.
The version for weak ascents/descents is A333213.
Cf. A064113, A066099, A233564, A333214, A333216.

b:= proc(n, h, t) option remember;
      if n=0 then `if`(t=0, 1, 0)
    elif t<0 then 0
    else add(b(n-j, j, `if`(j=h, t-1, t)), j=1..n)
      fi
    end:
T:= (n, k)-> b(n, -1, k):
seq(seq(T(n, k), k=0..n-1), n=1..15); # Alois P. Heinz, Oct 23 2011

b[n_, h_, t_] := b[n, h, t] = If[n == 0, If[t == 0, 1, 0], If[t<0, 0, Sum[b[n-j, j, If [j == h, t-1, t]], {j, 1, n}]]]; T[n_, k_] := b[n, -1, k]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Feb 20 2015, after Alois P. Heinz *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],n==0||Length[Split[#,#1!=#2&]]==k+1&]],{n,0,12},{k,0,n}] (* Gus Wiseman, Mar 23 2020 *)

A064113 Indices k such that (1/3)*(prime(k)+prime(k+1)+prime(k+2)) is a prime.

Original entry on oeis.org

2, 15, 36, 39, 46, 54, 55, 73, 102, 107, 110, 118, 129, 160, 164, 184, 187, 194, 199, 218, 239, 271, 272, 291, 339, 358, 387, 419, 426, 464, 465, 508, 520, 553, 599, 605, 621, 629, 633, 667, 682, 683, 702, 709, 710, 733, 761, 791, 813, 821, 822, 829, 830

Offset: 1

Jason Earls, Sep 08 2001

n such that d(n) = d(n+1), where d(n) = prime(n+1) - prime(n) = A001223(n).
Of interest because when I generalize it to d(n) = d(n+2), d(n) = d(n+3), etc. I am unable to find any positive number k such that d(n) = d(n+k) has no solution.
From Lei Zhou, Dec 06 2005: (Start)
When (1/3)*(prime(k) + prime(k+1) + prime(k+2)) is prime, then it is equal to prime(k+1).
Also, indices k such that (prime(k)+prime(k+2))/2 = prime(k+1).
The Mathematica program is based on the alternative definition. (End)
Inflection and undulation points of the primes, i.e., positions of zeros in A036263, the second differences of the primes. - Gus Wiseman, Mar 24 2020

			a(2) = 15 because (p(15)+p(16)+p(17)) = 1/3(47 + 53 + 59) = 53 (prime average of three successive primes).
Splitting the prime gaps into anti-runs gives: (1,2), (2,4,2,4,2,4,6,2,6,4,2,4,6), (6,2,6,4,2,6,4,6,8,4,2,4,2,4,14,4,6,2,10,2,6), (6,4,6), ... Then a(n) is the n-th partial sum of the lengths of these anti-runs. - _Gus Wiseman_, Mar 24 2020

Harry J. Smith, Table of n, a(n) for n=1..1000
Wikipedia, Inflection point

Indices of zeros in A036263 (second differences of primes).
Indices (A000720 = primepi) of balanced primes A006562, minus 1.
Cf. A001223, A024675, A075540, A075541.
Cf. A262138.
Complement of A333214.
First differences are A333216.
The version for strict ascents is A258025.
The version for strict descents is A258026.
The version for weak ascents is A333230.
The version for weak descents is A333231.
A triangle for anti-runs of compositions is A106356.
Lengths of maximal runs of prime gaps are A333254.
Anti-runs of compositions in standard order are A333381.
Cf. A000040, A084758, A124762, A124767, A238279.

import Data.List (elemIndices)
a064113 n = a064113_list !! (n-1)
a064113_list = map (+ 1) $ elemIndices 0 a036263_list
-- Reinhard Zumkeller, Jan 20 2012

ct = 0; Do[If[(Prime[k] + Prime[k + 2] - 2*Prime[k + 1]) == 0, ct++; n[ct] = k], {k, 1, 2000}]; Table[n[k], {k, 1, ct}] (* Lei Zhou, Dec 06 2005 *)
Join@@Position[Differences[Array[Prime,100],2],0] (* Gus Wiseman, Mar 24 2020 *)

d(n) = prime(n+1)-prime(n); j=[]; for(n=1,1500, if(d(n)==d(n+1), j=concat(j,n))); j

{ n=0; for (m=1, 10^9, if (d(m)==d(m+1), write("b064113.txt", n++, " ", m); if (n==1000, break)) ) } \\ Using d(n) above. - Harry J. Smith, Sep 07 2009

[n | n<-[1..888], !A036263(n)] \\ M. F. Hasler, Oct 15 2024

\\ More efficient for larges range of n:
A064113_upto(N, n=1, L=List(), q=prime(n+1), d=q-prime(n))={forprime(p=1+q,, if(d==d=p-q, listput(L,n); #LM. F. Hasler, Oct 15 2024

from itertools import count, islice
from sympy import prime, nextprime
def A064113_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)
A064113_list = list(islice(A064113_gen(),20)) # Chai Wah Wu, Feb 27 2024

A036263(a(n)) = 0; A122535(n) = A000040(a(n)); A006562(n) = A000040(a(n) + 1); A181424(n) = A000040(a(n) + 2). - Reinhard Zumkeller, Jan 20 2012
A262138(2*a(n)) = 0. - Reinhard Zumkeller, Sep 12 2015
a(n) = A000720(A006562(n)) - 1, where A000720 = (prime)pi, A006562 = balanced primes. - M. F. Hasler, Oct 15 2024

A333254 Lengths of maximal runs in the sequence of prime gaps (A001223).

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Mar 20 2020

nonn

Prime gaps are differences between adjacent prime numbers.

Also lengths of maximal arithmetic progressions of consecutive primes.

			The prime gaps split into the following runs: (1), (2,2), (4), (2), (4), (2), (4), (6), (2), (6), (4), (2), (4), (6,6), (2), (6), (4), ...

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Arithmetic progression
Wikipedia, Longest increasing subsequence

The version for A000002 is A000002. Similarly for A001462.
The unequal version is A333216.
The weakly decreasing version is A333212.
The weakly increasing version is A333215.
The strictly decreasing version is A333252.
The strictly increasing version is A333253.
Positions of first appearances are A335406.
The first term of the first length-n arithmetic progression of consecutive primes is A006560(n), with index A089180(n).
Prime gaps are A001223.
Positions of adjacent equal prime gaps are A064113.
Positions of adjacent unequal prime gaps are A333214.
Cf. A000040, A031217, A054800, A059044, A084758, A090832, A124767, A238279, A295235.

p:= 3: t:= 1: R:= NULL: s:= 1: count:= 0:
for i from 2 while count < 100 do
  q:= nextprime(p);
  g:= q-p; p:= q;
  if g = t then s:= s+1
  else count:= count+1; R:= R, s; t:= g; s:= 1;
  fi
od:
R; # Robert Israel, Jan 06 2021

Length/@Split[Differences[Array[Prime,100]],#1==#2&]//Most

Partial sums are A333214.

A257993 Least gap in the partition having Heinz number n; index of the least prime not dividing n.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3

Offset: 1

Emeric Deutsch, May 18 2015

nonn

The "least gap" of a partition is the least positive integer that is not a part of the partition. For example, the least gap of the partition [7,4,2,2,1] is 3.
We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436.
In the Maple program the subprogram B yields the partition with Heinz number n.
Sum of least gaps of all partitions of m = A022567(m).
From Antti Karttunen, Aug 22 2016: (Start)
Index of the least prime not dividing n. (After a formula given by Heinz.)
Least k such that A002110(k) does not divide n.
One more than the number of trailing zeros in primorial base representation of n, A049345.
(End)
The least gap is also called the mex (minimal excludant) of the partition. - Gus Wiseman, Apr 20 2021

			a(18) = 3 because the partition having Heinz number 18 = 2*3*3 is [1,2,2], having least gap equal to 3.

G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004, Cambridge.
Miklós Bóna, A Walk Through Combinatorics, World Scientific Publishing Co., 2002.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
P. J. Grabner and A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J., 12, 2006, 439-454.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics).
Index entries for sequences related to primorial base.

Cf. A215366, A002110, A022567, A049345, A053669, A055396, A064648, A276086, A276094, A276152.
Positions of 1's are A005408.
Positions of 2's are A047235.
The number of gaps is A079067.
The version for crank is A257989.
The triangle counting partitions by this statistic is A264401.
One more than A276084.
The version for greatest difference is A286469 or A286470.
A maximal instead of minimal version is A339662.
Positions of even terms are A342050.
Positions of odd terms are A342051.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709 counts partitions by sum and least difference.
A333214 lists positions of adjacent unequal prime gaps.
A339737 counts partitions by sum and greatest gap.
Cf. A001223, A001511, A005117, A026794, A029707, A072233, A079068, A098743, A124010, A279945, A325351, A325352.

with(numtheory): a := proc (n) local B, q: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: for q while member(q, B(n)) = true do  end do: q end proc: seq(a(n), n = 1 .. 150);
# second Maple program:
a:= n-> `if`(n=1, 1, (s-> min({$1..(max(s)+1)} minus s))(
        {map(x-> numtheory[pi](x[1]), ifactors(n)[2])[]})):
seq(a(n), n=1..100);  # Alois P. Heinz, May 09 2016
# faster:
A257993 := proc(n) local p, c; c := 1; p := 2;
while n mod p = 0 do p := nextprime(p); c := c + 1 od: c end:
seq(A257993(n), n=1..100); # Peter Luschny, Jun 04 2017

A053669[n_] := For[p = 2, True, p = NextPrime[p], If[CoprimeQ[p, n], Return[p]]]; a[n_] := PrimePi[A053669[n]]; Array[a, 100] (* Jean-François Alcover, Nov 28 2016 *)
Table[k = 1; While[! CoprimeQ[Prime@ k, n], k++]; k, {n, 100}] (* Michael De Vlieger, Jun 22 2017 *)

a(n) = forprime(p=2,, if (n % p, return(primepi(p)))); \\ Michel Marcus, Jun 22 2017

from sympy import nextprime, primepi
def a053669(n):
    p = 2
    while True:
        if n%p!=0: return p
        else: p=nextprime(p)
def a(n): return primepi(a053669(n)) # Indranil Ghosh, May 12 2017

(define (A257993 n) (let loop ((n n) (i 1)) (let* ((p (A000040 i)) (d (modulo n p))) (if (not (zero? d)) i (loop (/ (- n d) p) (+ 1 i))))))
;; Antti Karttunen, Aug 22 2016

a(n) = A000720(A053669(n)). - Alois P. Heinz, May 18 2015
From Antti Karttunen, Aug 22-30 2016: (Start)
a(n) = 1 + A276084(n).
a(n) = A055396(A276086(n)).
A276152(n) = A002110(a(n)).
(End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=1} 1/A002110(k) = 1.705230... (1 + A064648). - Amiram Eldar, Jul 23 2022
a(n) << log n/log log n. - Charles R Greathouse IV, Dec 03 2022

A simpler description added to the name by Antti Karttunen, Aug 22 2016

A054819 First term of weak prime quartet: p(m+1)-p(m) < p(m+2)-p(m+1) < p(m+3)-p(m+2).

Original entry on oeis.org

17, 41, 79, 107, 227, 281, 311, 347, 349, 379, 397, 439, 461, 499, 569, 641, 673, 677, 827, 857, 881, 907, 1031, 1061, 1091, 1187, 1229, 1277, 1301, 1319, 1367, 1427, 1429, 1451, 1487, 1489, 1549, 1607, 1619, 1621, 1697, 1877, 1997, 2027, 2087, 2153

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

			From _Gus Wiseman_, May 31 2020: (Start)
The first 10 strictly increasing prime gap quartets:
   17   19   23   29
   41   43   47   53
   79   83   89   97
  107  109  113  127
  227  229  233  239
  281  283  293  307
  311  313  317  331
  347  349  353  359
  349  353  359  367
  379  383  389  397
(End)

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A051635, A054800-A054840.
Prime gaps are A001223.
Second prime gaps are A036263.
Strictly decreasing prime gap quartets are A335278.
Strictly increasing prime gap quartets are A335277.
Equal prime gap quartets are A090832.
Weakly increasing prime gap quartets are A333383.
Weakly decreasing prime gap quartets are A333488.
Unequal prime gap quartets are A333490.
Partially unequal prime gap quartets are A333491.
Positions of adjacent equal prime gaps are A064113.
Positions of strict ascents in prime gaps are A258025.
Positions of strict descents in prime gaps are A258026.
Positions of adjacent unequal prime gaps are A333214.
Positions of weak ascents in prime gaps are A333230.
Positions of weak descents in prime gaps are A333231.
Lengths of maximal weakly decreasing sequences of prime gaps are A333212.
Lengths of maximal strictly increasing sequences of prime gaps are A333253.
Cf. A000040, A006560, A031217, A059044, A084758, A089180, A333253.

wpqQ[lst_]:=Module[{diffs=Differences[lst]},diffs[[1]]Harvey P. Dale, Jun 12 2012 *)
ReplaceList[Array[Prime,100],{_,x_,y_,z_,t_,_}/;y-xx] (* Gus Wiseman, May 31 2020 *)

a(n) = prime(A335277(n)). - Gus Wiseman, May 31 2020

A258025 Numbers k such that prime(k+2) - 2*prime(k+1) + prime(k) > 0.

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 13, 14, 17, 20, 22, 23, 26, 28, 29, 31, 33, 35, 38, 41, 43, 45, 49, 50, 52, 57, 60, 61, 64, 65, 67, 69, 70, 71, 75, 76, 78, 79, 81, 83, 85, 86, 89, 90, 93, 95, 96, 98, 100, 104, 105, 109, 113, 116, 117, 120, 122, 123, 124, 126, 131, 134

Offset: 1

Clark Kimberling, Jun 02 2015

			5 - 2*3 + 2 = 1, so a(1) = 5.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Partition of the positive integers:  A064113, A258025, A258026;
Corresponding partition of the primes: A063535, A063535, A147812.
Adjacent terms differing by 1 correspond to weak prime quartets A054819.
The version for the Kolakoski sequence is A156243.
The version for strict descents is A258026.
The version for weak ascents is A333230.
The version for weak descents is A333231.
First differences are A333212 (if the first term is 0).
Prime gaps are A001223.
Positions of adjacent equal prime gaps are A064113.
Weakly decreasing runs of compositions in standard order are A124765.
A triangle counting compositions by strict ascents is A238343.
Positions of adjacent unequal prime gaps are A333214.
Lengths of maximal anti-runs of prime gaps are A333216.
Cf. A000040, A036263, A084758, A114994, A124760, A124761, A124766, A333215.

u = Table[Sign[Prime[n+2] - 2 Prime[n+1] + Prime[n]], {n, 3, 200}];
Flatten[Position[u, 0]]   (* A064113 *)
Flatten[Position[u, 1]]   (* A258025 *)
Flatten[Position[u, -1]]  (* A258026 *)
Accumulate[Length/@Split[Differences[Array[Prime,100]],#1>=#2&]]//Most (* Gus Wiseman, Mar 25 2020 *)
Position[Partition[Prime[Range[150]],3,1],?(#[[3]]-2#[[2]]+#[[1]]> 0&),1,Heads->False]//Flatten (* _Harvey P. Dale, Dec 25 2021 *)

isok(k) = prime(k+2) - 2*prime(k+1) + prime(k) > 0; \\ Michel Marcus, Jun 03 2015

is(n,p=prime(n))=my(q=nextprime(p+1),r=nextprime(q+1)); p + r > 2*q
v=List(); n=0; forprime(p=2,1e4, if(is(n++,p), listput(v,n))); v \\ Charles R Greathouse IV, Jun 03 2015

from itertools import count, islice
from sympy import prime, nextprime
def A258025_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)
A258025_list = list(islice(A258025_gen(),20)) # Chai Wah Wu, Feb 27 2024

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

Original entry on oeis.org

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

Clark Kimberling, Jun 05 2015

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

			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

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

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.

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 *)

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

A054804 First term of strong prime quartets: prime(m+1)-prime(m) > prime(m+2)-prime(m+1) > prime(m+3)-prime(m+2).

Original entry on oeis.org

31, 61, 89, 211, 271, 293, 449, 467, 607, 619, 709, 743, 839, 863, 919, 1069, 1291, 1409, 1439, 1459, 1531, 1637, 1657, 1669, 1723, 1759, 1777, 1831, 1847, 1861, 1979, 1987, 2039, 2131, 2311, 2357, 2371, 2447, 2459, 2477, 2503, 2521, 2557, 2593, 2633

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

Primes preceding the first member of pairs of consecutive primes in A051634 ("strong primes"), see example. (A051634 lists the middle member of the triplets, here we list the first member of the quadruplets.) - M. F. Hasler, Oct 27 2018, corrected thanks to Gus Wiseman, Jun 01 2020.

			The first 10 strictly decreasing prime gap quartets:
   31  37  41  43
   61  67  71  73
   89  97 101 103
  211 223 227 229
  271 277 281 283
  293 307 311 313
  449 457 461 463
  467 479 487 491
  607 613 617 619
  619 631 641 643
For example, the primes (211,223,227,229) have differences (12,4,2), which are strictly decreasing, so 211 is in the sequence.
The second and third term of each quadruplet are consecutive terms in A051634: this is a characteristic property of this sequence. - _M. F. Hasler_, Jun 01 2020

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A051634, A054800-A054840.
Prime gaps are A001223.
Second prime gaps are A036263.
All of the following use prime indices rather than the primes themselves:
- Strictly decreasing prime gap quartets are A335278.
- Strictly increasing prime gap quartets are A335277.
- Equal prime gap quartets are A090832.
- Weakly increasing prime gap quartets are A333383.
- Weakly decreasing prime gap quartets are A333488.
- Unequal prime gap quartets are A333490.
- Partially unequal prime gap quartets are A333491.
- Adjacent equal prime gaps are A064113.
- Strict ascents in prime gaps are A258025.
- Strict descents in prime gaps are A258026.
- Adjacent unequal prime gaps are A333214.
- Weak ascents in prime gaps are A333230.
- Weak descents in prime gaps are A333231.
Maximal weakly increasing intervals of prime gaps are A333215.
Maximal strictly decreasing intervals of prime gaps are A333252.
Cf. also A000040, A006560, A031217, A054800, A054805, A054806, A054807, A059044, A084758, A089180, A333253, A335278.

primes:= select(isprime,[seq(i,i=3..10000,2)]):
L:=  primes[2..-1]-primes[1..-2]:
primes[select(t -> L[t+2] < L[t+1] and L[t+1] < L[t], [$1..nops(L)-2])]; # Robert Israel, Jun 28 2018

ReplaceList[Array[Prime,100],{_,x_,y_,z_,t_,_}/;y-x>z-y>t-z:>x] (* Gus Wiseman, May 31 2020 *)
Select[Partition[Prime[Range[400]],4,1],Max[Differences[#,2]]<0&][[All,1]] (* Harvey P. Dale, Jan 12 2023 *)

a(n) = prime(A335278(n)). - Gus Wiseman, May 31 2020

A333230 Positions of weak ascents in the sequence of differences between primes.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 10, 13, 14, 15, 17, 20, 22, 23, 26, 28, 29, 31, 33, 35, 36, 38, 39, 41, 43, 45, 46, 49, 50, 52, 54, 55, 57, 60, 61, 64, 65, 67, 69, 70, 71, 73, 75, 76, 78, 79, 81, 83, 85, 86, 89, 90, 93, 95, 96, 98, 100, 102, 104, 105, 107, 109, 110, 113

Offset: 1

Gus Wiseman, Mar 18 2020

nonn

Partial sums of A333252.

			The prime gaps split into the following strictly decreasing 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), ...

The version for the Kolakoski sequence is A022297.
The version for equal differences is A064113.
The version for strict ascents is A258025.
The version for strict descents is A258026.
The version for distinct differences is A333214.
The version for weak descents is A333231.
First differences are A333252 (if the first term is 0).
Prime gaps are A001223.
Weakly decreasing runs of standard compositions are counted by A124765.
Weakly increasing runs of standard compositions are counted by A124766.
Strictly increasing runs of standard compositions are counted by A124768.
Strictly decreasing runs of standard compositions are counted by A124769.
Runs of prime gaps with nonzero differences are A333216.
Cf. A000040, A000720, A036263, A054819, A084758, A333212, A333213, A333215.

Accumulate[Length/@Split[Differences[Array[Prime,100]],#1>#2&]]//Most
- or -
Select[Range[100],Prime[#+1]-Prime[#]<=Prime[#+2]-Prime[#+1]&]

Numbers k such that prime(k+2) - 2*prime(k+1) + prime(k) >= 0.

cp's OEIS Frontend

A238279 Triangle read by rows: T(n,k) is the number of compositions of n into nonzero parts with k parts directly followed by a different part, n>=0, 0<=k<=A004523(n-1).

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A106356 Triangle T(n,k) 0<=k

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A064113 Indices k such that (1/3)*(prime(k)+prime(k+1)+prime(k+2)) is a prime.

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A333254 Lengths of maximal runs in the sequence of prime gaps (A001223).

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A257993 Least gap in the partition having Heinz number n; index of the least prime not dividing n.

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A054819 First term of weak prime quartet: p(m+1)-p(m) < p(m+2)-p(m+1) < p(m+3)-p(m+2).

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