A084758 -id:A084758 - cp's OEIS frontend

A376682 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the noncomposite numbers (A008578).

1, 2, 1, 3, 1, 0, 5, 2, 1, 1, 7, 2, 0, -1, -2, 11, 4, 2, 2, 3, 5, 13, 2, -2, -4, -6, -9, -14, 17, 4, 2, 4, 8, 14, 23, 37, 19, 2, -2, -4, -8, -16, -30, -53, -90, 23, 4, 2, 4, 8, 16, 32, 62, 115, 205, 29, 6, 2, 0, -4, -12, -28, -60, -122, -237, -442, 31, 2, -4, -6, -6, -2, 10, 38, 98, 220, 457, 899

Offset: 0

Gus Wiseman, Oct 15 2024

Row k is the k-th differences of the noncomposite numbers.

			Array begins:
         n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:  n=9:
  -----------------------------------------------------------
  k=0:    1     2     3     5     7    11    13    17    19
  k=1:    1     1     2     2     4     2     4     2     4
  k=2:    0     1     0     2    -2     2    -2     2     2
  k=3:    1    -1     2    -4     4    -4     4     0    -6
  k=4:   -2     3    -6     8    -8     8    -4    -6    14
  k=5:    5    -9    14   -16    16   -12    -2    20   -28
  k=6:  -14    23   -30    32   -28    10    22   -48    48
  k=7:   37   -53    62   -60    38    12   -70    96   -70
  k=8:  -90   115  -122    98   -26   -82   166  -166    86
  k=9:  205  -237   220  -124   -56   248  -332   252   -86
Triangle begins:
    1
    2    1
    3    1    0
    5    2    1    1
    7    2    0   -1   -2
   11    4    2    2    3    5
   13    2   -2   -4   -6   -9  -14
   17    4    2    4    8   14   23   37
   19    2   -2   -4   -8  -16  -30  -53  -90
   23    4    2    4    8   16   32   62  115  205
   29    6    2    0   -4  -12  -28  -60 -122 -237 -442
   31    2   -4   -6   -6   -2   10   38   98  220  457  899

The version for modern primes (A000040) is A095195.
Initial rows: A008578, A075526, A036263 with 0 prepended.
Column n = 1 is A030016 (modern A007442).
A version for partitions is A175804, cf. A053445, A281425, A320590.
Antidiagonal-sums are A376683 (modern A140119), absolute A376684 (modern A376681).
First position of 0 is A376855 (modern A376678).
For composite instead of prime we have A377033.
For squarefree instead of prime we have A377038, nonsquarefree A377046.
For prime-power instead of composite we have A377051.
A000040 lists the primes, differences A001223, second A036263.
Cf. A002808, A064113, A065890, A084758, A173390, A233671, A258025, A258026, A333254, A376602 (zeros), A376651 (positives), A376652 (negatives), A376680, A377037.

nn=12;
t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1,!PrimeQ[#]&]&,1,2*nn],k],nn],{k,0,nn}]
(* or *)
nn=12;
q=Table[If[n==0,1,Prime[n]],{n,0,2nn}];
Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[i+k]],{k,0,j}],{j,0,nn},{i,nn}]

A(i,j) = Sum_{k=0..j} (-1)^(j-k) binomial(j,k) A008578(i+k).

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

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), ...

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 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.

A333216 Lengths of maximal subsequences without adjacent equal terms in the sequence of prime gaps.

Original entry on oeis.org

2, 13, 21, 3, 7, 8, 1, 18, 29, 5, 3, 8, 11, 31, 4, 20, 3, 7, 5, 19, 21, 32, 1, 19, 48, 19, 29, 32, 7, 38, 1, 43, 12, 33, 46, 6, 16, 8, 4, 34, 15, 1, 19, 7, 1, 23, 28, 30, 22, 8, 1, 7, 1, 52, 14, 56, 10, 26, 2, 30, 65, 5, 71, 12, 44, 39, 37, 6, 19, 47, 11, 10

Offset: 1

Gus Wiseman, Mar 15 2020

Prime gaps are differences between adjacent prime numbers.

Essentially the same as A145024. - R. J. Mathar, Mar 16 2020

			The prime gaps split into the following subsequences without adjacent equal terms: (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), (6,2,10,2,4,2,12), (12,4,2,4,6,2,10,6), ...

First differences of A064113.
The version for the Kolakoski sequence is A306323.
The weakly decreasing version is A333212.
The weakly increasing version is A333215.
The strictly decreasing version is A333252.
The strictly increasing version is A333253.
The equal version is A333254.
Cf. A000040, A001223, A084758, A106356, A124762, A124767, A333214.

Length/@Split[Differences[Array[Prime,100]],UnsameQ]//Most

Ones correspond to balanced prime quartets (A054800), so the sum of terms up to but not including the n-th one is A000720(A054800(n - 1)) = A090832(n).

A377050 Position of first appearance of zero in the n-th differences of the nonsquarefree numbers, or 0 if it does not appear.

Original entry on oeis.org

0, 0, 5, 11, 4, 129, 10, 89, 16, 161, 72, 77325, 71, 4870, 70, 253, 75, 737923, 166, 1648316, 165, 8753803, 164, 208366710, 163, 99489971, 162, 49493333, 161

Offset: 0

Gus Wiseman, Oct 19 2024

If a(29) is not 0, then it is > 10^12. - Lucas A. Brown, Oct 25 2024

			The fourth differences of A013929 begin: -6, -2, 5, 0, -7, 9, -6, 6, -7, ... so a(4) = 4.

Lucas A. Brown, Python program.

The version for primes is A376678, noncomposites A376855, composites A377037.
For squarefree instead of nonsquarefree numbers we have A377042.
For antidiagonal-sums we have A377047, absolute A377048.
For leading column we have A377049.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
Cf. A000961, A007674, A053797, A053806, A061398, A072284, A084758, A112925, A120992, A376311, A376591, A377051.

nn=10000;
u=Table[Differences[Select[Range[nn],!SquareFreeQ[#]&],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(17)-a(28) from Lucas A. Brown, Oct 25 2024

A333215 Lengths of maximal weakly increasing subsequences in the sequence of prime gaps (A001223).

Original entry on oeis.org

4, 2, 3, 2, 1, 4, 2, 1, 2, 3, 1, 2, 3, 2, 2, 3, 3, 2, 2, 3, 1, 3, 2, 3, 2, 1, 3, 1, 3, 2, 4, 2, 3, 3, 2, 2, 3, 1, 3, 1, 2, 3, 2, 2, 2, 3, 2, 3, 1, 2, 1, 4, 2, 4, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 3, 1, 3, 1, 3, 3, 1, 4, 4, 2, 2, 2, 3, 2, 3, 1, 5, 3, 2, 2, 4, 3, 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 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), ...

Prime gaps are A001223.
Ones correspond to strong prime quartets A054804.
Weakly increasing runs of compositions in standard order are A124766.
First differences of A258026 (with zero prepended).
The version for the Kolakoski sequence is A332875.
The weakly decreasing version is A333212.
The unequal version is A333216.
Positions of weak ascents in prime gaps are A333230.
The strictly decreasing version is A333252.
The strictly increasing version is A333253.
The equal version is A333254.
Cf. A000040, A000720, A036263, A054819, A064113, A084758, A124765, A124768, A258025, A333213, A333214.

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

Ones correspond to strong prime quartets (A054804), so the sum of terms up to but not including the n-th one is A000720(A054804(n - 1)).

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

Original entry on oeis.org

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.

A376681 Row sums of the absolute value of the array A095195(n, k) = n-th term of the k-th differences of the prime numbers (A000040).

Original entry on oeis.org

2, 4, 8, 10, 22, 36, 72, 134, 266, 500, 874, 1418, 2044, 2736, 4626, 15176, 41460, 95286, 196368, 372808, 660134, 1092790, 1682198, 2384724, 3147706, 4526812, 11037090, 36046768, 93563398, 214796426, 452129242, 885186658, 1619323680, 2763448574, 4368014812

Offset: 1

Gus Wiseman, Oct 15 2024

nonn

			The fourth row of A095195 is: (7, 2, 0, -1), so a(4) = 10.

For firsts instead of row-sums we have A007442 (modern version of A030016).
This is the absolute version of A140119.
If 1 is considered prime (A008578) we get A376684, absolute version of A376683.
For first zero-positions we have A376678 (modern version of A376855).
For composite instead of prime we have A377035.
For squarefree instead of prime we have A377040, nonsquarefree A377048.
A000040 lists the modern primes, differences A001223, seconds A036263.
A008578 lists the noncomposites, differences A075526, seconds A036263 with 0 prepended.
Cf. A064113, A065890, A084758, A173390, A233671, A258025, A333214, A333254, A376602, A376651, A376652, A377033.

nn=15;
t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1,!PrimeQ[#]&]&,2,2*nn],k],nn],{k,0,nn}]
Total/@Abs/@Table[t[[j,i-j+1]],{i,nn},{j,i}]

More terms from Pontus von Brömssen, Oct 17 2024

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

Original entry on oeis.org

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

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)).

cp's OEIS Frontend

A376682 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the noncomposite numbers (A008578).

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

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A333216 Lengths of maximal subsequences without adjacent equal terms in the sequence of prime gaps.

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A377050 Position of first appearance of zero in the n-th differences of the nonsquarefree numbers, or 0 if it does not appear.

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A333215 Lengths of maximal weakly increasing subsequences in the sequence of prime gaps (A001223).

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

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A376681 Row sums of the absolute value of the array A095195(n, k) = n-th term of the k-th differences of the prime numbers (A000040).

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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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