A375929 -id:A375929 - cp's OEIS frontend

A073445 Second differences of A002808, the sequence of composites.

0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0

Offset: 1

Labos Elemer, Aug 01 2002

			From _Gus Wiseman_, Oct 10 2024: (Start)
The composite numbers (A002808) are:
  4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
  2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, ...
with first differences (A073445):
  0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, ...
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Also first differences of A054546.
For first differences we had A073783 (ones A375929), run-lengths A376680.
Positions of zeros are A376602.
Positions of nonzeros are A376603.
Positions of ones are A376651, negative-ones A376652.
A002808 lists the composite numbers.
A064113 lists positions of adjacent equal prime gaps.
A333254 gives run-lengths of differences between consecutive primes.
Other second differences: A036263 (prime), A376590 (squarefree), A376596 (prime-power), A376604 (Kolakoski).
Cf. A076259, A174965, A251092, A376562, A376593, A376599.

a073445 n = a073445_list !! (n-1)
a073445_list = zipWith (-) (tail a073783_list) a073783_list
-- Reinhard Zumkeller, Jan 10 2013

c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x]; Table[c[w+2]-2*c[w+1]+c[w], {w, 200}]
(* second program *)
Differences[Select[Range[100],CompositeQ],2] (* Gus Wiseman, Oct 08 2024 *)

from sympy import primepi
def A073445(n):
    def iterfun(f,n=0):
        m, k = n, f(n)
        while m != k: m, k = k, f(k)
        return m
    return (a:=iterfun(f:=lambda x:n+primepi(x)+1,n))-((b:=iterfun(lambda x:f(x)+1,a))<<1)+iterfun(lambda x:f(x)+2,b) # Chai Wah Wu, Oct 03 2024

a(n) = c(n+2)-2*c(n+1)+c(n), where c(n) = A002808(n).

a(n) = A073783(n+1) - A073783(n). - Reinhard Zumkeller, Jan 10 2013

A065310 Number of occurrences of n-th prime in A065308, where A065308(j) = prime(j - pi(j)).

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Oct 29 2001

nonn

Seems identical to A054546. Each odd prime arises once or twice!?

First differences of A018252 (positive nonprime numbers). Including 0 gives A054546. Removing 1 gives A073783. - Gus Wiseman, Sep 15 2024

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

Cf. A000040, A000720, A054576, A065308-A065309.
For twin 2's see A169643.
Positions of 1's are A375926, complement A014689 (except first term).
Other families of numbers and their first-differences:
For prime numbers (A000040) we have A001223.
For composite numbers (A002808) we have A073783.
For nonprime numbers (A018252) we have A065310 (this).
For perfect powers (A001597) we have A053289.
For non-perfect-powers (A007916) we have A375706.
For squarefree numbers (A005117) we have A076259.
For nonsquarefree numbers (A013929) we have A078147.
For prime-powers inclusive (A000961) we have A057820.
For prime-powers exclusive (A246655) we have A057820(>1).
For non-prime-powers inclusive (A024619) we have A375735.
For non-prime-powers exclusive (A361102) we have A375708.
Cf. A046933, A053707, A054546, A110969, A176246, A251092, A373403, A375929.

t=Table[Prime[w-PrimePi[w]], {w, a, b}] Table[Count[t, Prime[n]], {n, c, d}]
Differences[Select[Range[100],!PrimeQ[#]&]] (* Gus Wiseman, Sep 15 2024 *)

{ p=1; f=2; m=1; for (n=1, 1000, a=0; p=nextprime(p + 1); while (p==f, a++; m++; f=prime(m - primepi(m))); write("b065310.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 16 2009

A375927 Numbers k such that A005117(k+1) - A005117(k) = 1. In other words, the k-th squarefree number is 1 less than the next.

Original entry on oeis.org

1, 2, 4, 5, 7, 9, 10, 14, 15, 18, 19, 21, 22, 24, 25, 27, 28, 30, 35, 36, 38, 40, 41, 43, 44, 46, 48, 49, 51, 53, 54, 58, 59, 62, 63, 65, 66, 68, 69, 71, 72, 74, 76, 79, 80, 82, 84, 85, 87, 88, 90, 94, 96, 97, 101, 102, 105, 107, 108, 110, 111, 113, 114, 116

Offset: 1

Gus Wiseman, Sep 12 2024

nonn

The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^2-1)) = 0.53071182... (A065469). - Amiram Eldar, Sep 15 2024

			The squarefree numbers are 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ... which first increase by one after terms 1, 2, 4, 5, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Positions of 1's in A076259.
For prime-powers (A246655) we have A375734.
First differences are A373127.
For nonsquarefree instead of squarefree we have A375709.
For nonprime numbers we have A375926, differences A373403.
For composite numbers we have A375929.
The complement is A375930, differences A120992.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A007674, A061399, A065469, A072284, A110969, A294242, A373409, A373410, A373573.

Join@@Position[Differences[Select[Range[100],SquareFreeQ[#]&]],1]

lista(kmax) = {my(is1 = 1, is2, c = 1); for(k = 2, kmax, is2 = issquarefree(k); if(is2, c++); if(is1 && is2, print1(c-1, ", ")); is1 = is2);} \\ Amiram Eldar, Sep 15 2024

A376602 Inflection and undulation points in the sequence of composite numbers (A002808).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 14, 15, 16, 18, 20, 21, 22, 25, 27, 29, 32, 33, 34, 37, 38, 39, 41, 43, 44, 45, 48, 50, 52, 53, 54, 57, 60, 61, 62, 65, 66, 67, 68, 69, 72, 74, 76, 78, 80, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 96, 99, 100, 101, 103, 105, 106, 107, 108

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A073445) are zero.

			The composite numbers (A002808) are:
  4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
  2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, ...
with first differences (A073445):
  0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, ...
with zeros at (A376602):
  1, 3, 5, 7, 9, 11, 14, 15, 16, 18, 20, 21, 22, 25, 27, 29, 32, 33, 34, 37, 38, ...

Wikipedia, Inflection point
Gus Wiseman, Inflection and undulation points in the sequence of composite numbers.

Partitions into composite numbers are counted by A023895, factorizations A050370.
For prime instead of composite we have A064113.
These are the positions of zeros in A073445.
For first differences we had A073783, ones A375929, complement A065890.
For concavity in primes we have A258025/A258026, weak A333230/A333231.
For upward concavity (instead of zero) we have A376651, downward A376652.
The complement is A376603.
For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376603 (nonzero curvature), A376651 (concave-up), A376652 (concave-down).
For inflection and undulation points: A064113 (prime), A376588 (non-perfect-power), A376591 (squarefree), A376594 (nonsquarefree), A376597 (prime-power), A376600 (non-prime-power).
Cf. A000040, A036263, A251092, A333214, A333216, A333254.

Join@@Position[Differences[Select[Range[100],CompositeQ],2],0]

A376603 Points of nonzero curvature in the sequence of composite numbers (A002808).

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 13, 17, 19, 23, 24, 26, 28, 30, 31, 35, 36, 40, 42, 46, 47, 49, 51, 55, 56, 58, 59, 63, 64, 70, 71, 73, 75, 77, 79, 81, 82, 94, 95, 97, 98, 102, 104, 112, 114, 118, 119, 123, 124, 126, 127, 131, 132, 136, 138, 146, 148, 150, 152, 162, 163

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A073445) are nonzero.

			The composite numbers (A002808) are:
  4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
  2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, ...
with first differences (A073445):
  0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, ...
with nonzero terms at (A376603):
  2, 4, 6, 8, 10, 12, 13, 17, 19, 23, 24, 26, 28, 30, 31, 35, 36, 40, 42, 46, 47, ...

Dominic McCarty, Table of n, a(n) for n = 1..1000
Gus Wiseman, Points of nonzero curvature in the sequence of composite numbers.

Partitions into composite numbers are counted by A023895, factorizations A050370.
These are the positions of nonzero terms in A073445.
For first differences we had A073783, ones A375929, complement A065890.
For prime instead of composite we have A333214.
The complement is A376602.
For upward concavity (instead of nonzero) we have A376651, downward A376652.
For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376602 (zeros), A376651 (concave-up), A376652 (concave-down).
For nonzero curvature: A333214 (prime), A376589 (non-perfect-power), A376592 (squarefree), A376595 (nonsquarefree), A376598 (prime-power), A376601 (non-prime-power).
Cf. A000961, A064113, A246655, A251092, A258025, A333254.

Join@@Position[Sign[Differences[Select[Range[100],CompositeQ],2]],1|-1]

A376651 Points of upward concavity in the sequence of composite numbers (A002808).

Original entry on oeis.org

4, 8, 12, 17, 23, 26, 30, 35, 40, 46, 49, 55, 58, 63, 70, 73, 77, 81, 94, 97, 102, 112, 118, 123, 126, 131, 136, 146, 150, 162, 173, 176, 180, 185, 195, 200, 205, 210, 216, 219, 229, 242, 245, 249, 262, 267, 276, 280, 285, 292, 297, 302, 305, 310, 317, 320

Offset: 1

Gus Wiseman, Oct 06 2024

nonn

These are points at which the second differences (A073445) are positive.

Also positions of strict ascents in the first differences (A073783) of composite numbers (A002808).

			The composite numbers are (A002808):
  4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
  2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, ...
with first differences (A073445):
  0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, ...
with positive terms at (A376651):
  4, 8, 12, 17, 23, 26, 30, 35, 40, 46, 49, 55, 58, 63, 70, 73, 77, 81, 94, 97, ...

Dominic McCarty, Table of n, a(n) for n = 1..1000
Gus Wiseman, Points of upward concavity in the sequence of composite numbers.

The version for A000002 is A022297, negative A156242.
Partitions into composite numbers are counted by A023895, factorizations A050370.
For first differences we had A065310 or A073783, ones A375929.
These are the positions of positive terms in A073445, negative A376652.
For prime instead of composite we have A258025, negative A258026.
For zero second differences (instead of positive) we have A376602.
For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376602 (inflections and undulations), A376603 (nonzero curvature), A376652 (concave-down).
Cf. A000961, A057820, A064113, A065890, A251092, A333254, A376604.

Join@@Position[Sign[Differences[Select[Range[1000],CompositeQ],2]],1]

A376652 Points of downward concavity in the sequence of composite numbers (A002808).

Original entry on oeis.org

2, 6, 10, 13, 19, 24, 28, 31, 36, 42, 47, 51, 56, 59, 64, 71, 75, 79, 82, 95, 98, 104, 114, 119, 124, 127, 132, 138, 148, 152, 163, 174, 178, 181, 187, 196, 201, 206, 212, 217, 221, 230, 243, 247, 250, 263, 268, 278, 281, 286, 293, 298, 303, 306, 311, 318, 321

Offset: 1

Gus Wiseman, Oct 06 2024

nonn

These are points at which the second differences (A073445) are negative.

Also positions of strict descents in the first differences (A073783) of composite numbers (A002808).

			The composite numbers are (A002808):
  4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
  2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, ...
with second differences (A073445):
  0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, ...
with negative terms at (A376651):
  2, 6, 10, 13, 19, 24, 28, 31, 36, 42, 47, 51, 56, 59, 64, 71, 75, 79, 82, 95, 98, ...

Robert Israel, Table of n, a(n) for n = 1..10000
Gus Wiseman, Points of downward concavity in the sequence of composite numbers.

The version for A000002 is A156242, positive A022297.
Partitions into composite numbers are counted by A023895, factorizations A050370.
For first differences we had A065310 or A073783, ones A375929.
These are the positions of negative terms in A073445, positive A376651.
For prime instead of composite we have A258026, positive A258025.
For zero second differences instead of negative we have A376602.
For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376602 (inflections and undulations), A376603 (nonzero curvature), A376651 (concave-up).
Cf. A000961, A007916, A057820, A064113, A065890, A251092, A333254, A376604.

Comps:= remove(isprime, [seq(i,i=4..1000)]):
D1:= Comps[2..-1]-Comps[1..-2]:
D2:= D1[2..-1]-D1[1..-2]:
select(t -> D2[t] < 0, [$1..nops(D2)]); # Robert Israel, Nov 06 2024

Join@@Position[Sign[Differences[Select[Range[1000],CompositeQ],2]],-1]

A376680 Run-lengths of first differences of composite numbers.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 1, 4, 2, 4, 1, 2, 2, 2, 1, 4, 1, 4, 2, 4, 1, 2, 2, 4, 1, 2, 1, 4, 1, 6, 1, 2, 2, 2, 2, 2, 1, 12, 1, 2, 1, 4, 2, 8, 2, 4, 1, 4, 1, 2, 1, 4, 1, 4, 2, 8, 2, 2, 2, 10, 1, 10, 1, 2, 2, 2, 1, 4, 2, 8, 1, 4, 1, 4, 1, 4, 2, 4, 1, 2, 2, 8, 1, 12, 1, 2

Offset: 1

Gus Wiseman, Oct 10 2024

nonn

Also first differences of A376603 (points of nonzero curvature in the composite numbers).

			The composite numbers (A002808) are:
  4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
  2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, ...
with runs:
  (2,2), (1,1), (2,2), (1,1), (2,2), (1,1), (2), (1,1,1,1), (2,2), (1,1,1,1), ...
with lengths (A376680):
  2, 2, 2, 2, 2, 2, 1, 4, 2, 4, 1, 2, 2, 2, 1, 4, 1, 4, 2, 4, 1, 2, 2, 4, 1, 2, ...

These are the run-lengths of A073783, ones A375929.
For prime instead of composite we have A333254, first appearances A335406.
These are the first differences of A376603.
A000040 lists the prime numbers, first differences A001223, second differences A036263.
A002808 lists the composite numbers, differences A073783.
A064113 lists positions of adjacent equal prime gaps.
A073445 gives second differences of composite numbers, zeros A376602.
Cf. A054546, A076259, A174965, A251092, A376651, A376652.

Length/@Split[Differences[Select[Range[100],CompositeQ]]]

A375926 Numbers k such that A018252(k+1) = A018252(k) + 1. In other words, the k-th nonprime number is 1 less than the next.

Original entry on oeis.org

4, 5, 8, 9, 12, 13, 15, 16, 17, 18, 21, 22, 23, 24, 26, 27, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 44, 45, 46, 47, 49, 50, 53, 54, 55, 56, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 73, 74, 77, 78, 81, 82, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95

Offset: 1

Gus Wiseman, Sep 11 2024

nonn

			The nonprime numbers are 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ... which increase by 1 after term 4, term 5, term 8, etc.

The complement appears to be A014689, except the first term.
Positions of 1's in A065310 (see also A054546, A073783).
First differences are A373403 (except first).
The version for non-prime-powers is A375713, differences A373672.
The version for prime-powers is A375734, differences A373671.
The version for non-perfect-powers is A375740.
The version for composite numbers is A375929.
A000040 lists the prime numbers, differences A001223.
A018252 lists the nonprimes, exclusive A002808.
A046933 counts composite numbers between primes.
Cf. A000961, A006549, A057820, A176246, A246655, A251092, A375708.

Join@@Position[Differences[Select[Range[100],!PrimeQ[#]&]],1]

from sympy import primepi
def A375926(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+bisection(lambda y:primepi(x+1+y))-1
    return bisection(f,n,n) # Chai Wah Wu, Sep 15 2024

A375930 Numbers k such that A005117(k+1) - A005117(k) > 1. In other words, the k-th squarefree number is more than 1 less than the next.

Original entry on oeis.org

3, 6, 8, 11, 12, 13, 16, 17, 20, 23, 26, 29, 31, 32, 33, 34, 37, 39, 42, 45, 47, 50, 52, 55, 56, 57, 60, 61, 64, 67, 70, 73, 75, 77, 78, 81, 83, 86, 89, 91, 92, 93, 95, 98, 99, 100, 103, 104, 106, 109, 112, 115, 117, 120, 121, 122, 125, 127, 130, 133, 136, 139

Offset: 1

Gus Wiseman, Sep 12 2024

nonn

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/(p^2-1)) = 1 - A065469 = 0.46928817... . - Amiram Eldar, Sep 15 2024

			The squarefree numbers are 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ... which first increase by more than one after positions 3, 6, 8, 11, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

For nonprime numbers: A014689, complement A375926, differences A373403.
For composite numbers: A065890 shifted, complement A375929.
Positions of terms > 1 in A076259.
First differences are A120992, complement A373127.
The complement is A375927.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
Cf. A007674, A061399, A065469, A068781, A072284, A110969, A294242, A375709.

Join@@Position[Differences[Select[Range[100],SquareFreeQ[#]&]],_?(#>1&)]

lista(kmax) = {my(is1 = 1, is2, c = 1); for(k = 2, kmax, is2 = issquarefree(k); if(is2, c++); if(is1 && !is2, print1(c, ", ")); is1 = is2);} \\ Amiram Eldar, Sep 15 2024

cp's OEIS Frontend

A073445 Second differences of A002808, the sequence of composites.

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A065310 Number of occurrences of n-th prime in A065308, where A065308(j) = prime(j - pi(j)).

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A375927 Numbers k such that A005117(k+1) - A005117(k) = 1. In other words, the k-th squarefree number is 1 less than the next.

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A376602 Inflection and undulation points in the sequence of composite numbers (A002808).

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A376603 Points of nonzero curvature in the sequence of composite numbers (A002808).

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A376651 Points of upward concavity in the sequence of composite numbers (A002808).

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A376652 Points of downward concavity in the sequence of composite numbers (A002808).

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A376680 Run-lengths of first differences of composite numbers.

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