A333254 -id:A333254 - cp's OEIS frontend

A376678 Position of first zero in the n-th differences of the primes, or 0 if it does not appear.

0, 0, 2, 7, 69, 13, 47, 58, 9, 43, 3553, 100, 7019, 14082, 68097, 14526, 149677, 2697, 481054, 979719, 631894, 29811, 25340978, 50574254, 7510843, 210829337, 67248861, 224076286, 910615647, 931510269, 452499644, 2880203722, 396680865, 57954439970, 77572822440, 35394938648

Offset: 0

Gus Wiseman, Oct 14 2024

nonn

Do the k-th differences of the primes contain a zero for all k > 1?

			The third differences of the primes begin:
  -1, 2, -4, 4, -4, 4, 0, -6, 8, ...
so a(3) = 7.

Lucas A. Brown, Python program.

If 1 is considered prime (A008578) we get A376855.
The zeros of second differences are A064113, complement A333214.
This is the position at which 0 first appears in row n of A095195.
For composite instead of prime we have A377037.
For squarefree instead of prime we have A377042, nonsquarefree A377050.
For prime-power instead of prime we have A377055.
A000040 lists the primes, first differences A001223, second A036263.
Cf. A000720, A007442, A030016, A065890, A084758, A140119, A258025, A258026, A333254, A349643, A376681, A376682, A376683.

nn=100000;
u=Table[Differences[Select[Range[nn],PrimeQ],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]]]}]

a(n) = A000720(A349643(n)) for n >= 2. - Pontus von Brömssen, Oct 17 2024

a(17)-a(32) from Pontus von Brömssen, Oct 17 2024

a(33)-a(35) from Lucas A. Brown, Nov 03 2024

A376684 Antidiagonal-sums of the absolute value of the array A376682(n,k) = n-th term of the k-th differences of the noncomposite numbers (A008578).

Original entry on oeis.org

1, 3, 4, 9, 12, 27, 50, 109, 224, 471, 942, 1773, 3118, 4957, 7038, 9373, 16256, 55461, 150622, 346763, 718972, 1377101, 2462220, 4114987, 6387718, 9112455, 12051830, 17160117, 40946860, 134463917, 349105370, 800713921, 1684145408, 3297536923, 6040907554

Offset: 0

Gus Wiseman, Oct 15 2024

nonn

			The fourth antidiagonal of A376682 is: (7, 2, 0, -1, -2), so a(4) = 12.

For the modern primes (A000040) we have A376681, absolute version of A140119.
For firsts instead of row-sums we have A030016, modern A007442.
These are the antidiagonal-sums of the absolute value of A376682 (modern A095195).
This is the absolute version of A376683.
For first zero-positions we have A376855, modern A376678.
A000040 lists the modern primes, differences A001223, seconds A036263.
A008578 lists the noncomposites, first differences A075526.
Cf. A002808, A064113, A065890, A173390, A233671, A258025, A333214, A333254, A376680, A377033, A377037.

nn=12;
t=Table[Take[Differences[NestList[NestWhile[#+1&,#+1,!PrimeQ[#]&]&,1,2*nn],k],nn],{k,0,nn}];
Total/@Table[Abs[t[[j,i-j+1]]],{i,nn},{j,i}]

A376855 Position of first 0 in the n-th differences of the noncomposite numbers (A008578), or 0 if it does not appear.

Original entry on oeis.org

0, 0, 1, 8, 70, 14, 48, 59, 10, 44, 3554, 101, 7020, 14083, 68098, 14527, 149678, 2698, 481055, 979720, 631895, 29812, 25340979, 50574255, 7510844, 210829338, 67248862, 224076287, 910615648, 931510270, 452499645, 2880203723, 396680866, 57954439971, 77572822441, 35394938649

Offset: 0

Gus Wiseman, Oct 15 2024

nonn

			The third differences of the noncomposite numbers begin: 1, -1, 2, -4, 4, -4, 4, 0, -6, 8, ... so a(3) = 8.

Lucas A. Brown, Python program.

For firsts instead of positions of zeros we have A030016, modern A007442.
These are the first zero-positions in A376682, modern A376678.
For row-sums instead of zero-positions we have A376683, modern A140119.
For absolute row-sums we have A376684, modern A376681.
For composite instead of noncomposite we have A377037.
For squarefree instead of noncomposite we have A377042, nonsquarefree A377050.
For prime-power instead of noncomposite we have A377055.
A000040 lists the modern primes, differences A001223, seconds A036263.
A008578 lists the noncomposite numbers, first differences A075526.
Cf. A002808, A064113, A065890, A084758, A173390, A258025, A333214, A333215, A333254, A377033.

nn=10000;
u=Table[Differences[Select[Range[nn],#==1||PrimeQ[#]&],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]]]}]

a(16)-a(21) from Alois P. Heinz, Oct 18 2024

a(22)-a(35) from Lucas A. Brown, Nov 03 2024

A333212 Lengths of maximal weakly decreasing subsequences in the sequence of prime gaps (A001223).

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 3, 1, 3, 3, 2, 1, 3, 2, 1, 2, 2, 2, 3, 3, 2, 2, 4, 1, 2, 5, 3, 1, 3, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 2, 2, 1, 3, 1, 3, 2, 1, 2, 2, 4, 1, 4, 4, 3, 1, 3, 2, 1, 1, 2, 5, 3, 2, 2, 2, 2, 2, 1, 3, 1, 3, 1, 2, 1, 3, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 3

Offset: 1

Gus Wiseman, Mar 14 2020

nonn

Prime gaps are differences between adjacent prime numbers.

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

First differences of A258025 (with zero prepended).
The version for the Kolakoski sequence is A332273.
The weakly increasing version is A333215.
The unequal version is A333216.
The strictly decreasing version is A333252.
The strictly increasing version is A333253.
The equal version is A333254.
Prime gaps are A001223.
Positions of adjacent equal differences are A064113.
Weakly decreasing runs of compositions in standard order are A124765.
Positions of strict ascents in the sequence of prime gaps are A258025.
Cf. A000040, A000720, A001221, A036263, A054819, A084758, A114994, A124760, A124761, A124768, A333213, A333214.

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

Ones correspond to weak prime quartets A054819, so the sum of terms up to but not including the n-th one is A000720(A054819(n - 1)).

A333253 Lengths of maximal strictly increasing subsequences in the sequence of prime gaps (A001223).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 18 2020

nonn

Prime gaps are differences between adjacent prime numbers.

			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), ...

Wikipedia, Longest increasing subsequence

The weakly decreasing version is A333212.
The weakly increasing version is A333215.
The unequal version is A333216.
First differences of A333231 (if its first term is 0).
The strictly decreasing version is A333252.
The equal version is A333254.
Prime gaps are A001223.
Strictly increasing runs of compositions in standard order are A124768.
Positions of strict ascents in the sequence of prime gaps are A258025.
Cf. A000040, A054819, A064113, A084758, A333214, A333230, A333255.

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

Partial sums are A333231. The partial sum up to but not including the n-th one is A333382(n).

A373820 Run-lengths (differing by 0) of antirun-lengths (differing by > 2) of odd primes.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 22 2024

nonn

Run-lengths of the version of A027833 with 1 prepended.

			The antiruns of odd primes (differing by > 2) begin:
   3
   5
   7  11
  13  17
  19  23  29
  31  37  41
  43  47  53  59
  61  67  71
  73  79  83  89  97 101
 103 107
 109 113 127 131 137
 139 149
 151 157 163 167 173 179
 181 191
 193 197
 199 211 223 227
 229 233 239
 241 251 257 263 269
 271 277 281
with lengths:
1, 1, 2, 2, 3, 3, 4, 3, 6, 2, 5, 2, 6, 2, 2, ...
with runs:
  1  1
  2  2
  3  3
  4
  3
  6
  2
  5
  2
  6
  2  2
  4
  3
  5
  3
  4
with lengths a(n).

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

Run-lengths of A027833 (if we prepend 1), partial sums A029707.
For runs we have A373819, run-lengths of A251092.
Positions of first appearances are A373827, sorted A373826.
A000040 lists the primes.
A001223 gives differences of consecutive primes, run-lengths A333254, run-lengths of run-lengths A373821.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
For prime runs: A001359, A006512, A025584, A067774, A373405, A373406.
For composite runs: A005381, A054265, A068780, A373403, A373404.
Cf. A005117, A006560, A006562, A037201, A038664, A049579, A073051, A076259, A122535, A176246.

Length/@Split[Length/@Split[Select[Range[3,1000],PrimeQ],#2-#1>2&]//Most]//Most

A376592 Points of nonzero curvature in the sequence of squarefree numbers (A005117).

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 13, 15, 17, 19, 20, 22, 23, 25, 26, 28, 29, 30, 31, 34, 36, 37, 38, 39, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 54, 57, 59, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 85, 86, 88, 89, 90, 91, 92, 93, 94, 95

Offset: 1

Gus Wiseman, Oct 04 2024

nonn

These are points at which the second differences (A376590) are nonzero.

			The squarefree numbers (A005117) are:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, ...
with first differences (A076259):
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, ...
with first differences (A376590):
  0, 1, -1, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 2, 0, -2, 0, 1, -1, 0, 1, -1, 0, 1, ...
with nonzeros at (A376591):
  2, 3, 5, 6, 7, 8, 10, 13, 15, 17, 19, 20, 22, 23, 25, 26, 28, 29, 30, 31, 34, 36, ...

Gus Wiseman, Points of nonzero curvature in the squarefree numbers.

The first differences were A076259, see also A375927, A376305, A376306, A376307, A376311.
These are the nonzeros of A376590.
The complement is A376591.
A000040 lists the prime numbers, differences A001223.
A005117 lists squarefree numbers, complement A013929 (differences A078147).
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
For points of nonzero curvature: A333214 (prime), A376603 (composite), A376589 (non-perfect-power), A376595 (nonsquarefree), A376598 (prime-power), A376601 (non-prime-power).
For squarefree numbers: A076259 (first differences), A376590 (second differences), A376591 (inflection and undulation points).
Cf. A007674, A036263, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A333254, A373198, A376655.

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

A377035 Antidiagonal-sums of the absolute value 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, 12, 14, 18, 21, 28, 34, 40, 47, 74, 96, 110, 138, 286, 715, 2393, 8200, 25731, 72468, 184716, 431575, 934511, 1892267, 3605315, 6494464, 11116110, 18134549, 28348908, 42701927, 62290660, 88313069, 120999433, 159769475, 221775851, 483797879

Offset: 1

Gus Wiseman, Oct 18 2024

nonn

			The fourth antidiagonal of A377033 is (9, 1, -1, -1), so a(4) = 12.

The version for prime instead of composite is A376681, absolute version of A140119.
The version for noncomposite is A376684, absolute version of A376683.
This is the antidiagonal-sums of absolute value of the array A377033.
For squarefree instead of composite we have A377040, absolute version of A377039.
For nonsquarefree instead of composite we have A377048, absolute version of A377047.
For prime-power instead of composite we have A377053, absolute version of A377052.
Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).
A000040 lists the primes, differences A001223, seconds A036263.
A002808 lists the composite numbers, differences A073783, seconds 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[120],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[Abs[t[[j,i-j+1]]],{i,Length[q]/2},{j,i}]

A373822 Sum of the n-th maximal run of first differences of odd primes.

Original entry on oeis.org

4, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 12, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 12, 4, 12, 2, 10, 2, 4, 2, 24, 4, 2, 4, 6, 2, 10, 18, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 12, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4

Offset: 1

Gus Wiseman, Jun 22 2024

nonn

Run-sums of A001223. For run-lengths instead of run-sums we have A333254.

			The odd primes are
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with first differences
2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, ...
with runs
(2,2), (4), (2), (4), (2), (4), (6), (2), (6), (4), (2), (4), (6,6), ...
with sums a(n).

Run-sums of A001223.
For run-lengths we have A333254, run-lengths of run-lengths A373821.
Dividing by two gives A373823.
A000040 lists the primes.
A027833 gives antirun lengths of odd primes (partial sums A029707).
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
A373820 gives run-lengths of antirun-lengths of odd primes.
For prime runs: A001359, A006512, A025584, A067774, A373405, A373406.
Cf. A037201, A038664, A054265, A176246, A251092, A373404, A373825.

Total/@Split[Differences[Select[Range[3,1000],PrimeQ]]]

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A376678 Position of first zero in the n-th differences of the primes, or 0 if it does not appear.

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A376684 Antidiagonal-sums of the absolute value of the array A376682(n,k) = n-th term of the k-th differences of the noncomposite numbers (A008578).

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A376855 Position of first 0 in the n-th differences of the noncomposite numbers (A008578), or 0 if it does not appear.

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A333212 Lengths of maximal weakly decreasing subsequences in the sequence of prime gaps (A001223).

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A333253 Lengths of maximal strictly increasing subsequences in the sequence of prime gaps (A001223).

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A373820 Run-lengths (differing by 0) of antirun-lengths (differing by > 2) of odd primes.

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A376592 Points of nonzero curvature in the sequence of squarefree numbers (A005117).

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A376680 Run-lengths of first differences of composite numbers.

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