A025505 -id:A025505 - cp's OEIS frontend

A333231 Positions of weak descents in the sequence of differences between primes.

2, 4, 6, 9, 11, 12, 15, 16, 18, 19, 21, 24, 25, 27, 30, 32, 34, 36, 37, 39, 40, 42, 44, 46, 47, 48, 51, 53, 54, 55, 56, 58, 59, 62, 63, 66, 68, 72, 73, 74, 77, 80, 82, 84, 87, 88, 91, 92, 94, 97, 99, 101, 102, 103, 106, 107, 108, 110, 111, 112, 114, 115, 118

Offset: 1

Gus Wiseman, Mar 18 2020

Partial sums of A333253.

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

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Thomas Bloom, Let d_n = p_(n+1)-p_n. The set of n such that d_(n+1) >= d_n has density 1/2, and similarly for d_(n+1) <= d_n. Furthermore, there are infinitely many n such that d_(n+1) = d_n, Erdős Problems.
Terence Tao, Erdős problem database, see no. 218.

The version for the Kolakoski sequence is A025505.
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 ascents is A333230.
First differences are A333253 (if the first term is 0).
Prime gaps are A001223.
Weakly decreasing runs of compositions in standard order are A124765.
Strictly increasing runs of compositions in standard order are A124768.
Runs of prime gaps with nonzero differences are A333216.
Cf. A000040, A000720, A001221, A036263, A054819, A084758, A114994, A124760, A124761, A124766, A124769, A333212.

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.

A376560 Points of upward concavity in the sequence of perfect-powers (A001597). Positives of A376559.

Original entry on oeis.org

1, 3, 4, 6, 8, 9, 10, 11, 12, 15, 16, 17, 20, 22, 23, 26, 27, 28, 31, 32, 33, 34, 35, 36, 37, 38, 41, 42, 43, 44, 46, 47, 48, 49, 50, 53, 54, 55, 57, 58, 60, 61, 62, 63, 64, 65, 67, 68, 69, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91

Offset: 1

Gus Wiseman, Sep 30 2024

nonn

These are points at which the second differences are positive.
Perfect-powers (A001597) are numbers with a proper integer root.
Note that, for some sources, upward concavity is negative curvature.

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, 33, ...
with first differences (A376559):
  1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, 2, ...
with positive positions (A376560):
  1, 3, 4, 6, 8, 9, 10, 11, 12, 15, 16, 17, 20, 22, 23, 26, 27, 28, 31, 32, 33, 34, ...

Robert Israel, Table of n, a(n) for n = 1..10000
Gus Wiseman, Points of upward concavity in the perfect-powers.

The version for A000002 is A022297, complement A025505. See also A054354, A376604.
For first differences we have A053289, union A023055, firsts A376268, A376519.
For primes instead of perfect-powers we have A258025.
These are positions of positive terms in A376559.
For downward concavity we have A376561 (probably the complement).
A001597 lists the perfect-powers.
A064113 lists positions of adjacent equal prime gaps.
A333254 gives run-lengths of differences between consecutive primes.
Second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power), A376599 (non-prime-power).
Cf. A006549, A025475, A030173, A045542, A052410, A053707, A054819, A069623, A174965, A216765, A251092, A376308.

N:= 10^6: # to use perfect powers <= N
S:= {1,seq(seq(i^j,j=2..floor(log[i](N))),i=2..isqrt(N))}:
L:= sort(convert(S,list)):
DL:= L[2..-1]-L[1..-2]:
D2L:= DL[2..-1]-DL[1..-2]:
select(i -> D2L[i]>0, [$1..nops(D2L)]); # Robert Israel, Dec 01 2024

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Join@@Position[Sign[Differences[Select[Range[1000],perpowQ],2]],1]

A376561 Points of downward concavity in the sequence of perfect-powers (A001597).

Original entry on oeis.org

2, 5, 7, 13, 14, 18, 19, 21, 24, 25, 29, 30, 39, 40, 45, 51, 52, 56, 59, 66, 70, 71, 74, 87, 94, 101, 102, 108, 110, 112, 113, 119, 127, 135, 143, 144, 156, 157, 160, 161, 169, 178, 187, 196, 205, 206, 215, 224, 225, 234, 244, 263, 273, 283, 284, 293, 294, 304

Offset: 1

Gus Wiseman, Sep 30 2024

nonn

These are points at which the second differences are negative.
Perfect-powers (A001597) are numbers with a proper integer root.
Note that, for some sources, downward concavity is positive curvature.
From Robert Israel, Oct 31 2024: (Start)
The first case of two consecutive numbers in the sequence is a(4) = 13 and a(5) = 14.
The first case of three consecutive numbers is a(293) = 2735, a(294) = 2736, a(295) = 2737.
The first case of four consecutive numbers, if it exists, involves a(k) with k > 69755. (End)

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, 33, ...
with first differences (A376559):
  1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, 2, ...
with negative positions (A376561):
  2, 5, 7, 13, 14, 18, 19, 21, 24, 25, 29, 30, 39, 40, 45, 51, 52, 56, 59, 66, 70, ...

Robert Israel, Table of n, a(n) for n = 1..10000
Gus Wiseman, Points of downward concavity in the perfect-powers.

The version for A000002 is A025505, complement A022297. See also A054354, A376604.
For first differences we have A053289, union A023055, firsts A376268, A376519.
For primes instead of perfect-powers we have A258026.
For upward concavity we have A376560 (probably the complement).
A000961 lists the prime-powers inclusive, exclusive A246655.
A001597 lists the perfect-powers.
A007916 lists the non-perfect-powers.
A112344 counts partitions into perfect-powers, factorizations A294068.
A333254 gives run-lengths of differences between consecutive primes.
Second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power), A376599 (non-prime-power).
Cf. A025475, A045542, A052410, A054819, A064113, A069623, A174965, A216765, A251092, A336416, A336417, A375702.

N:= 10^6: # to use perfect powers <= N
P:= {seq(seq(i^m,i=2..floor(N^(1/m))), m=2 .. ilog2(N))}: nP:= nops(P):
P:= sort(convert(P,list)):
select(i -> 2*P[i] > P[i-1]+P[i+1], [$2..nP-1]); # Robert Israel, Oct 31 2024

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Join@@Position[Sign[Differences[Select[Range[1000],perpowQ],2]],-1]

cp's OEIS Frontend

A333231 Positions of weak descents in the sequence of differences between primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A376560 Points of upward concavity in the sequence of perfect-powers (A001597). Positives of A376559.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A376561 Points of downward concavity in the sequence of perfect-powers (A001597).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica