A073445 -id:A073445 - cp's OEIS frontend

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

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]

A379301 Positive integers whose prime indices include a unique composite number.

Original entry on oeis.org

7, 13, 14, 19, 21, 23, 26, 28, 29, 35, 37, 38, 39, 42, 43, 46, 47, 52, 53, 56, 57, 58, 61, 63, 65, 69, 70, 71, 73, 74, 76, 77, 78, 79, 84, 86, 87, 89, 92, 94, 95, 97, 101, 103, 104, 105, 106, 107, 111, 112, 113, 114, 115, 116, 117, 119, 122, 126, 129, 130, 131

Offset: 1

Gus Wiseman, Dec 25 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 70 are {1,3,4}, so 70 is in the sequence.
The prime indices of 98 are {1,4,4}, so 98 is not in the sequence.

For no composite parts we have A302540, counted by A034891 (strict A036497).
For all composite parts we have A320629, counted by A023895 (strict A204389).
For a unique prime part we have A331915, counted by A379304 (strict A379305).
Positions of one in A379300.
Partitions of this type are counted by A379302 (strict A379303).
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A066247 is the characteristic function for the composite numbers.
A377033 gives k-th differences of composite numbers, see A073445, A377034-A377037.
Other counts of prime indices:
- A087436 postpositive, see A038550.
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
- A379311 old prime, see A379312-A379315.
Cf. A113646, A376680.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[Select[prix[#],CompositeQ]]==1&]

A377034 Antidiagonal-sums of the array A377033(n,k) = n-th term of the k-th differences of the composite numbers (A002808).

Original entry on oeis.org

4, 8, 10, 8, 14, 14, 11, 24, 10, 20, 37, -10, 56, 26, -52, 260, -659, 2393, -8128, 25703, -72318, 184486, -430901, 933125, -1888651, 3597261, -6479654, 11086964, -18096083, 28307672, -42644743, 62031050, -86466235, 110902085, -110907437, 52379, 483682985

Offset: 1

Gus Wiseman, Oct 17 2024

sign

Row-sums of the triangle version of A377033.

			The fourth antidiagonal of A377033 is (9, 1, -1, -1), so a(4) = 8.

The version for prime instead of composite is A140119, noncomposite A376683.
This is the antidiagonal-sums of the array A377033, absolute version A377035.
For squarefree instead of composite we have A377039, absolute version A377040.
For nonsquarefree instead of composite we have A377047, absolute version A377048.
For prime-power instead of composite we have A377052, absolute version A377053.
Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).
A000040 lists the primes, differences A001223, second A036263.
A002808 lists the composite numbers, differences A073783, second A073445.
A008578 lists the noncomposites, differences A075526.
Cf. A018252, A065310, A065890, A333254, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680, A377036.

q=Select[Range[100],CompositeQ];
t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[i+k]],{k,0,j}],{j,0,Length[q]/2},{i,Length[q]/2}];
Total/@Table[t[[j,i-j+1]],{i,Length[q]/2},{j,i}]

A377037 Position of first zero in the n-th differences of the composite numbers (A002808), or 0 if it does not appear.

Original entry on oeis.org

1, 14, 2, 65, 1, 83, 2, 7, 1, 83, 2, 424, 12, 32, 11, 733, 10, 940, 9, 1110, 8, 1110, 7, 1110, 6, 1110, 112, 1110, 111, 1110, 110, 2192, 109, 13852, 108, 13852, 107, 13852, 106, 13852, 105, 17384, 104, 17384, 103, 17384, 102, 17384, 101, 27144, 552, 28012, 551

Offset: 2

Gus Wiseman, Oct 17 2024

nonn

			The third differences of the composite numbers are:
  -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -2, 1, 0, 0, 1, -1, -1, ...
so a(3) = 14.

Alois P. Heinz, Table of n, a(n) for n = 2..100

The version for prime instead of composite is A376678.
For noncomposite numbers we have A376855.
This is the first position of 0 in row n of the array A377033.
For squarefree instead of composite we have A377042, nonsquarefree A377050.
For prime-power instead of composite we have A377055.
Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).
A000040 lists the primes, differences A001223, second A036263.
A002808 lists the composite numbers, differences A073783, second A073445.
A008578 lists the noncomposites, differences A075526.
A377036 gives first term of the n-th differences of the composite numbers, for primes A007442 or A030016.
Cf. A018252, A064113, A065310, A065890, A140119, A173390, A233671, A258025, A258026, A350004, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680, A377034, A377035.

nn=10000;
u=Table[Differences[Select[Range[nn],CompositeQ],k],{k,2,16}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
m=Table[Position[u[[k]],0][[1,1]],{k,mnrm[Union[First/@Position[u,0]]]}]

Offset 2 from Michel Marcus, Oct 18 2024

a(17)-a(54) from Alois P. Heinz, Oct 18 2024

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]

A379312 Positive integers whose prime indices include a unique 1 or prime number.

Original entry on oeis.org

2, 3, 5, 11, 14, 17, 21, 26, 31, 35, 38, 39, 41, 46, 57, 58, 59, 65, 67, 69, 74, 77, 83, 86, 87, 94, 95, 98, 106, 109, 111, 115, 119, 122, 127, 129, 141, 142, 143, 145, 146, 147, 157, 158, 159, 178, 179, 182, 183, 185, 191, 194, 202, 206, 209, 211, 213, 214

Offset: 1

Gus Wiseman, Dec 28 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
   11: {5}
   14: {1,4}
   17: {7}
   21: {2,4}
   26: {1,6}
   31: {11}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   41: {13}
   46: {1,9}
   57: {2,8}
   58: {1,10}
   59: {17}
   65: {3,6}
   67: {19}
   69: {2,9}
   74: {1,12}
   77: {4,5}

These "old" primes are listed by A008578.
For no composite parts we have A302540, counted by A034891 (strict A036497).
For all composite parts we have A320629, counted by A023895 (strict A204389).
For a unique prime part we have A331915, counted by A379304 (strict A379305).
Positions of ones in A379311, see A379313.
Partitions of this type are counted by A379314, strict A379315.
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A080339 is the characteristic function for the old prime numbers.
A376682 gives k-th differences of old prime numbers, see A030016, A075526.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
Cf. A038550, A073445, A087436, A173390, A379300, A379301, A379302.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],#==1||PrimeQ[#]&]]==1&]

A376604 Second differences of the Kolakoski sequence (A000002). First differences of A054354.

Original entry on oeis.org

-1, -1, 1, 1, -2, 2, -1, -1, 2, -1, -1, 1, 1, -2, 1, 1, -1, -1, 2, -2, 1, 1, -2, 2, -1, -1, 1, 1, -2, 1, 1, -2, 2, -1, -1, 2, -1, -1, 1, 1, -2, 2, -1, -1, 2, -2, 1, 1, -2, 1, 1, -1, -1, 2, -1, -1, 1, 1, -2, 2, -1, -1, 2, -1, -1, 1, 1, -2, 1, 1, -2, 2, -1, -1

Offset: 1

Gus Wiseman, Oct 02 2024

sign

Since A000002 has no runs of length 3, this sequence contains no zeros.

The densities appear to approach (1/3, 1/3, 1/6, 1/6).

			The Kolakoski sequence (A000002) is:
  1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, ...
with first differences (A054354):
  1, 0, -1, 0, 1, -1, 1, 0, -1, 1, 0, -1, 0, 1, -1, 0, 1, 0, -1, 1, -1, 0, 1, -1, ...
with first differences (A376604):
  -1, -1, 1, 1, -2, 2, -1, -1, 2, -1, -1, 1, 1, -2, 1, 1, -1, -1, 2, -2, 1, 1, -2, ...

A001462 is Golomb's sequence.
A078649 appears to be zeros of the first and third differences.
A288605 gives positions of first appearances of each balance.
A306323 gives a 'broken' version.
A333254 lists run-lengths of differences between consecutive primes.
For the Kolakoski sequence (A000002):
- Restrictions: A074264, A100428, A100429, A156263, A156264.
- Patterns: A013947, A013948, A022292, A074262, A074263, A156242, A156243.
- Statistics: A054353, A074286, A088568, A156077, A156253.
- Transformations: A054354, A156728, A332273, A332875, A333229, A376604.
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power), A376599 (non-prime-power).

kolagrow[q_]:=If[Length[q]<2,Take[{1,2},Length[q]+1],Append[q,Switch[{q[[Length[Split[q]]]],q[[-2]],Last[q]},{1,1,2},1,{1,2,1},2,{2,1,1},2,{2,1,2},2,{2,2,1},1,{2,2,2},1]]]
kol[n_]:=Nest[kolagrow,{1},n-1];
Differences[kol[100],2]

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]]]

cp's OEIS Frontend

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

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A379301 Positive integers whose prime indices include a unique composite number.

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A377034 Antidiagonal-sums of the array A377033(n,k) = n-th term of the k-th differences of the composite numbers (A002808).

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A377037 Position of first zero in the n-th differences of the composite numbers (A002808), or 0 if it does not appear.

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

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A379312 Positive integers whose prime indices include a unique 1 or prime number.

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A376604 Second differences of the Kolakoski sequence (A000002). First differences of A054354.

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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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