cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 22 results. Next

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

Views

Author

Gus Wiseman, Oct 17 2024

Keywords

Comments

Row-sums of the triangle version of A377033.

Examples

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

Crossrefs

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.

Programs

  • Mathematica
    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}]

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

Views

Author

Gus Wiseman, Oct 18 2024

Keywords

Examples

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

Crossrefs

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.

Programs

  • Mathematica
    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}]

A377051 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the powers of primes.

Original entry on oeis.org

1, 2, 1, 3, 1, 0, 4, 1, 0, 0, 5, 1, 0, 0, 0, 7, 2, 1, 1, 1, 1, 8, 1, -1, -2, -3, -4, -5, 9, 1, 0, 1, 3, 6, 10, 15, 11, 2, 1, 1, 0, -3, -9, -19, -34, 13, 2, 0, -1, -2, -2, 1, 10, 29, 63, 16, 3, 1, 1, 2, 4, 6, 5, -5, -34, -97, 17, 1, -2, -3, -4, -6, -10, -16, -21, -16, 18, 115
Offset: 0

Views

Author

Gus Wiseman, Oct 20 2024

Keywords

Comments

Row k of the array is the k-th differences of A000961.

Examples

			Array form:
        n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:  n=9:
  ----------------------------------------------------------
  k=0:   1     2     3     4     5     7     8     9    11
  k=1:   1     1     1     1     2     1     1     2     2
  k=2:   0     0     0     1    -1     0     1     0     1
  k=3:   0     0     1    -2     1     1    -1     1    -3
  k=4:   0     1    -3     3     0    -2     2    -4     6
  k=5:   1    -4     6    -3    -2     4    -6    10    -8
  k=6:  -5    10    -9     1     6   -10    16   -18     5
  k=7:  15   -19    10     5   -16    26   -34    23     9
  k=8: -34    29    -5   -21    42   -60    57   -14   -42
  k=9:  63   -34   -16    63  -102   117   -71   -28   104
Triangle form:
    1
    2    1
    3    1    0
    4    1    0    0
    5    1    0    0    0
    7    2    1    1    1    1
    8    1   -1   -2   -3   -4   -5
    9    1    0    1    3    6   10   15
   11    2    1    1    0   -3   -9  -19  -34
   13    2    0   -1   -2   -2    1   10   29   63
   16    3    1    1    2    4    6    5   -5  -34  -97

Crossrefs

Row k=0 is A000961, exclusive A246655.
Row k=1 is A057820.
Row k=2 is A376596.
The version for primes is A095195, noncomposites A376682, composites A377033.
A version for partitions is A175804, cf. A053445, A281425, A320590.
For squarefree numbers we have A377038, nonsquarefree A377046.
Triangle row-sums are A377052, absolute version A377053.
Column n = 1 is A377054, for primes A007442 or A030016.
First position of 0 in each row is A377055.
A000040 lists the primes, differences A001223, seconds A036263.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
Cf. A025475, A053707, A093555, A174965, A361102, A376340, A376598, A376653.

Programs

  • Mathematica
    nn=12;
t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1,!PrimePowerQ[#]&]&,1,2*nn],k],nn],{k,0,nn}]
Table[t[[j,i-j+1]],{i,nn},{j,i}]

Formula

A(i,j) = Sum_{k=0..j} (-1)^(j-k)*binomial(j,k)*A000961(i+k).

A095195 T(n,0) = prime(n), T(n,k) = T(n,k-1)-T(n-1,k-1), 0<=k

Original entry on oeis.org

2, 3, 1, 5, 2, 1, 7, 2, 0, -1, 11, 4, 2, 2, 3, 13, 2, -2, -4, -6, -9, 17, 4, 2, 4, 8, 14, 23, 19, 2, -2, -4, -8, -16, -30, -53, 23, 4, 2, 4, 8, 16, 32, 62, 115, 29, 6, 2, 0, -4, -12, -28, -60, -122, -237, 31, 2, -4, -6, -6, -2, 10, 38, 98, 220, 457, 37, 6, 4, 8, 14, 20, 22, 12
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 22 2004

Keywords

Comments

T(n,0)=A000040(n); T(n,1)=A001223(n-1) for n>1; T(n,2)=A036263(n-2) for n>2; T(n,n-1)=A007442(n) for n>1.
Row k of the array (not the triangle) is the k-th differences of the prime numbers. - Gus Wiseman, Jan 11 2025

Examples

			Triangle begins:
   2;
   3,  1;
   5,  2,  1;
   7,  2,  0, -1;
  11,  4,  2,  2,  3;
  13,  2, -2, -4, -6, -9;
Alternative: array form read by antidiagonals:
     2,   3,   5,   7,  11,  13,  17,  19,  23,  29,  31,...
     1,   2,   2,   4,   2,   4,   2,   4,   6,   2,   6,...
     1,   0,   2,  -2,   2,  -2,   2,   2,  -4,   4,  -2,...
    -1,   2,  -4,   4,  -4,   4,   0,  -6,   8,  -6,   0,...
     3,  -6,   8,  -8,   8,  -4,  -6,  14, -14,   6,   4,...
    -9,  14, -16,  16, -12,  -2,  20, -28,  20,  -2,  -8,...
    23, -30,  32, -28,  10,  22, -48,  48, -22,  -6,  10,..,
   -53,  62, -60,  38,  12, -70,  96, -70,  16,  16, -12,...
   115,-122,  98, -26, -82, 166,-166,  86,   0, -28,  28,...
  -237, 220,-124, -56, 248,-332, 252, -86, -28,  56, -98,...
   457,-344,  68, 304,-580, 584,-338,  58,  84,-154, 308,...

Links

Crossrefs

Cf. A036262-A036271.
Cf. A140119 (row sums).
Below, the inclusive primes (A008578) are 1 followed by A000040. See also A075526.
Rows of the array (columns of the triangle) begin: A000040, A001223, A036263.
Column n = 1 of the array is A007442, inclusive A030016.
The version for partition numbers is A175804, see A053445, A281425, A320590.
First position of 0 is A376678, inclusive A376855.
Absolute antidiagonal-sums are A376681, inclusive A376684.
The inclusive version is A376682.
For composite instead of prime we have A377033, see A377034-A377037.
For squarefree instead of prime we have A377038, nonsquarefree A377046.
Column n = 2 of the array is A379542.
Cf. A002808, A064113, A065890, A073783, A258026, A293467, A333254, A376683, A377051.

Programs

  • Haskell
    a095195 n k = a095195_tabl !! (n-1) !! (k-1)
a095195_row n = a095195_tabl !! (n-1)
a095195_tabl = f a000040_list [] where
   f (p:ps) xs = ys : f ps ys where ys = scanl (-) p xs
-- Reinhard Zumkeller, Oct 10 2013
  • Maple
    A095195A := proc(n,k) # array, k>=0, n>=0
    option remember;
    if n =0 then
        ithprime(k+1) ;
    else
        procname(n-1,k+1)-procname(n-1,k) ;
    end if;
end proc:
A095195 := proc(n,k) # triangle, 0<=k=1
        A095195A(k,n-k-1) ;
end proc: # R. J. Mathar, Sep 19 2013
  • Mathematica
    T[n_, 0] := Prime[n]; T[n_, k_] /; 0 <= k < n := T[n, k] = T[n, k-1] - T[n-1, k-1]; Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Feb 01 2017 *)
nn=6;
t=Table[Differences[Prime[Range[nn]],k],{k,0,nn}];
Table[t[[j,i-j+1]],{i,nn},{j,i}] (* Gus Wiseman, Jan 11 2025 *)

A175804 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the n-th term of the k-th differences of partition numbers A000041.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, -1, 0, 1, 3, 2, 1, 1, 2, 5, -4, -2, -1, 0, 2, 7, 9, 5, 3, 2, 2, 4, 11, -21, -12, -7, -4, -2, 0, 4, 15, 49, 28, 16, 9, 5, 3, 3, 7, 22, -112, -63, -35, -19, -10, -5, -2, 1, 8, 30, 249, 137, 74, 39, 20, 10, 5, 3, 4, 12, 42, -539, -290, -153, -79, -40, -20, -10, -5, -2, 2, 14, 56
Offset: 0

Views

Author

Alois P. Heinz, Dec 04 2010

Keywords

Comments

Odlyzko showed that the k-th differences of A000041(n) alternate in sign with increasing n up to a certain index n_0(k) and then stay positive.
Are there any zeros after the first four, which all lie in columns k = 1, 2? - Gus Wiseman, Dec 15 2024

Examples

			Square array A(n,k) begins:
   1,  0,  1, -1,  2,  -4,   9,  ...
   1,  1,  0,  1, -2,   5, -12,  ...
   2,  1,  1, -1,  3,  -7,  16,  ...
   3,  2,  0,  2, -4,   9, -19,  ...
   5,  2,  2, -2,  5, -10,  20,  ...
   7,  4,  0,  3, -5,  10, -20,  ...
  11,  4,  3, -2,  5, -10,  22,  ...

Links

Crossrefs

Columns k=0-5 give: A000041, A002865, A053445, A072380, A081094, A081095.
Main diagonal gives A379378.
Cf. A119712, A155861.
For primes we have A095195 or A376682.
Row n = 0 is A281425.
Row n = 1 is A320590 except first term.
For composites we have A377033.
For squarefree numbers we have A377038.
For nonsquarefree numbers we have A377046.
For prime powers we have A377051.
Antidiagonal sums are A377056, absolute value version A378621.
The version for strict partitions is A378622, first column A293467.
A000009 counts strict integer partitions, differences A087897, A378972.
Cf. A008284, A047966, A098859, A116608, A225486, A325242, A225485 or A325280, A377285, A378970.

Programs

  • Maple
    A41:= combinat[numbpart]:
DD:= proc(p) proc(n) option remember; p(n+1) -p(n) end end:
A:= (n,k)-> (DD@@k)(A41)(n):
seq(seq(A(n, d-n), n=0..d), d=0..11);
  • Mathematica
    max = 11; a41 = Array[PartitionsP, max+1, 0]; a[n_, k_] := Differences[a41, k][[n+1]]; Table[a[n, k-n], {k, 0, max}, {n, 0, k}] // Flatten (* Jean-François Alcover, Aug 29 2014 *)
nn=5;Table[Table[Sum[(-1)^(k-i)*Binomial[k,i]*PartitionsP[n+i],{i,0,k}],{k,0,nn}],{n,0,nn}] (* Gus Wiseman, Dec 15 2024 *)

Formula

A(n,k) = (Delta^(k) A000041)(n).
A(n,k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k,i) * A000041(n+i). In words, row x is the inverse zero-based binomial transform of A000041 shifted left x times. - Gus Wiseman, Dec 15 2024

A377046 Array read by downward antidiagonals where A(n,k) is the n-th term of the k-th differences of nonsquarefree numbers.

Original entry on oeis.org

4, 8, 4, 9, 1, -3, 12, 3, 2, 5, 16, 4, 1, -1, -6, 18, 2, -2, -3, -2, 4, 20, 2, 0, 2, 5, 7, 3, 24, 4, 2, 2, 0, -5, -12, -15, 25, 1, -3, -5, -7, -7, -2, 10, 25, 27, 2, 1, 4, 9, 16, 23, 25, 15, -10, 28, 1, -1, -2, -6, -15, -31, -54, -79, -94, -84, 32, 4, 3, 4, 6, 12, 27, 58, 112, 191, 285, 369
Offset: 0

Views

Author

Gus Wiseman, Oct 19 2024

Keywords

Comments

Row k is the k-th differences of A013929.

Examples

			Array form:
        n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:  n=9:
  ---------------------------------------------------------
  k=0:   4     8     9    12    16    18    20    24    25
  k=1:   4     1     3     4     2     2     4     1     2
  k=2:  -3     2     1    -2     0     2    -3     1    -1
  k=3:   5    -1    -3     2     2    -5     4    -2     4
  k=4:  -6    -2     5     0    -7     9    -6     6    -7
  k=5:   4     7    -5    -7    16   -15    12   -13    10
  k=6:   3   -12    -2    23   -31    27   -25    23   -13
  k=7: -15    10    25   -54    58   -52    48   -36    13
  k=8:  25    15   -79   112  -110   100   -84    49     1
  k=9: -10   -94   191  -222   210  -184   133   -48   -57
Triangle form:
   4
   8   4
   9   1  -3
  12   3   2   5
  16   4   1  -1  -6
  18   2  -2  -3  -2   4
  20   2   0   2   5   7   3
  24   4   2   2   0  -5 -12 -15
  25   1  -3  -5  -7  -7  -2  10  25
  27   2   1   4   9  16  23  25  15 -10
  28   1  -1  -2  -6 -15 -31 -54 -79 -94 -84
  32   4   3   4   6  12  27  58 112 191 285 369

Crossrefs

Initial rows: A013929, A078147, A376593.
The version for primes is A095195, noncomposites A376682, composites A377033.
A version for partitions is A175804, cf. A053445, A281425, A320590.
For squarefree numbers we have A377038, sums A377039, absolute A377040.
Triangle row-sums are A377047, absolute version A377048.
Column n = 1 is A377049, for squarefree A377041, for prime A007442 or A030016.
First position of 0 in each row is A377050.
For prime-power instead of nonsquarefree we have A377051.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
Cf. A000961, A007674, A053797, A053806, A061398, A072284, A112925, A120992, A376311, A376591, A376592.

Programs

  • Mathematica
    nn=9;
t=Table[Take[Differences[NestList[NestWhile[#+1&,#+1,SquareFreeQ[#]&]&,4,2*nn],k],nn],{k,0,nn}]
Table[t[[j,i-j+1]],{i,nn},{j,i}]

Formula

A(i,j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) A013929(i+k).

A377038 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the squarefree numbers.

Original entry on oeis.org

1, 2, 1, 3, 1, 0, 5, 2, 1, 1, 6, 1, -1, -2, -3, 7, 1, 0, 1, 3, 6, 10, 3, 2, 2, 1, -2, -8, 11, 1, -2, -4, -6, -7, -5, 3, 13, 2, 1, 3, 7, 13, 20, 25, 22, 14, 1, -1, -2, -5, -12, -25, -45, -70, -92, 15, 1, 0, 1, 3, 8, 20, 45, 90, 160, 252, 17, 2, 1, 1, 0, -3, -11, -31, -76, -166, -326, -578
Offset: 0

Views

Author

Gus Wiseman, Oct 18 2024

Keywords

Comments

Row n is the k-th differences of A005117 = the squarefree numbers.

Examples

			Array form:
        n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:  n=9:
  ----------------------------------------------------------
  k=0:   1     2     3     5     6     7    10    11    13
  k=1:   1     1     2     1     1     3     1     2     1
  k=2:   0     1    -1     0     2    -2     1    -1     0
  k=3:   1    -2     1     2    -4     3    -2     1     1
  k=4:  -3     3     1    -6     7    -5     3     0    -2
  k=5:   6    -2    -7    13   -12     8    -3    -2     3
  k=6:  -8    -5    20   -25    20   -11     1     5    -5
  k=7:   3    25   -45    45   -31    12     4   -10    10
  k=8:  22   -70    90   -76    43    -8   -14    20   -19
  k=9: -92   160  -166   119   -51    -6    34   -39    28
Triangle form:
   1
   2   1
   3   1   0
   5   2   1   1
   6   1  -1  -2  -3
   7   1   0   1   3   6
  10   3   2   2   1  -2  -8
  11   1  -2  -4  -6  -7  -5   3
  13   2   1   3   7  13  20  25  22
  14   1  -1  -2  -5 -12 -25 -45 -70 -92
  15   1   0   1   3   8  20  45  90 160 252

Crossrefs

Row k=0 is A005117.
Row k=1 is A076259.
Row k=2 is A376590.
The version for primes is A095195, noncomposites A376682, composites A377033.
A version for partitions is A175804, cf. A053445, A281425, A320590.
Triangle row-sums are A377039, absolute version A377040.
Column n = 1 is A377041, for primes A007442 or A030016.
First position of 0 in each row is A377042.
For nonsquarefree instead of squarefree numbers we have A377046.
For prime-powers instead of squarefree numbers we have A377051.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, complement A013929 (differences A078147).
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
Cf. A007674, A053797, A053806, A061398, A072284, A075526, A112925, A112926, A120992, A373198, A376311, A376591, A376592.

Programs

  • Mathematica
    nn=9;
t=Table[Take[Differences[NestList[NestWhile[#+1&,#+1,!SquareFreeQ[#]&]&,1,2*nn],k],nn],{k,0,nn}]
Table[t[[j,i-j+1]],{i,nn},{j,i}]

Formula

A(i,j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) A005117(i+k).

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).

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Oct 15 2024

Keywords

Comments

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

Examples

			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

Crossrefs

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.

Programs

  • Mathematica
    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}]

Formula

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

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

Views

Author

Gus Wiseman, Dec 25 2024

Keywords

Comments

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.

Examples

			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.

Crossrefs

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.

Programs

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

A376683 Antidiagonal-sums 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, 6, 27, -20, 109, -182, 471, -868, 1737, -2872, 4345, -4700, 1133, 14060, -55275, 150462, -346093, 717040, -1369351, 2432872, -4002905, 5964846, -7524917, 6123130, 4900199, -40900410, 134309057, -348584552, 798958881, -1678213106, 3277459119
Offset: 0

Views

Author

Gus Wiseman, Oct 15 2024

Keywords

Examples

			The fourth anti-diagonal of A376682 is: (7, 2, 0, -1, -2), so a(4) = 6.

Crossrefs

The modern version (for A000040 instead of A008578) is A140119.
The absolute version is A376681.
Antidiagonal-sums of A376682 (modern version A095195).
For composite instead of noncomposite we have A377033.
For squarefree instead of noncomposite we have A377038, nonsquarefree A377046.
A000040 lists the modern primes, differences A001223, second A036263.
A008578 lists the noncomposites, first differences A075526.
Cf. A007442, A030016, A064113, A065890, A002808, A173390, A233671, A333214, A333254, A376678, A376684, A376855.

Programs

  • Mathematica
    nn=12;
t=Table[Take[Differences[NestList[NestWhile[#+1&,#+1,!PrimeQ[#]&]&,1,2*nn],k],nn],{k,0,nn}];
Total/@Table[t[[j,i-j+1]],{i,nn},{j,i}]
Showing 1-10 of 22 results. Next

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