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.

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A378622 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the strict partition numbers A000009.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 2, 1, 1, 1, 2, 0, -1, -2, -3, 3, 1, 1, 2, 4, 7, 4, 1, 0, -1, -3, -7, -14, 5, 1, 0, 0, 1, 4, 11, 25, 6, 1, 0, 0, 0, -1, -5, -16, -41, 8, 2, 1, 1, 1, 1, 2, 7, 23, 64, 10, 2, 0, -1, -2, -3, -4, -6, -13, -36, -100, 12, 2, 0, 0, 1, 3, 6, 10, 16, 29, 65, 165
Offset: 0

Views

Author

Gus Wiseman, Dec 13 2024

Keywords

Examples

			As a table (read by antidiagonals downward):
        n=0:  n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:
  ----------------------------------------------------------
  k=0:   1     1     1     2     2     3     4     5     6
  k=1:   0     0     1     0     1     1     1     1     2
  k=2:   0     1    -1     1     0     0     0     1     0
  k=3:   1    -2     2    -1     0     0     1    -1     0
  k=4:  -3     4    -3     1     0     1    -2     1     1
  k=5:   7    -7     4    -1     1    -3     3     0    -3
  k=6: -14    11    -5     2    -4     6    -3    -3     7
  k=7:  25   -16     7    -6    10    -9     0    10   -14
  k=8: -41    23   -13    16   -19     9    10   -24    24
  k=9:  64   -36    29   -35    28     1   -34    48   -34
As a triangle (read by rows):
   1
   1   0
   1   0   0
   2   1   1   1
   2   0  -1  -2  -3
   3   1   1   2   4   7
   4   1   0  -1  -3  -7 -14
   5   1   0   0   1   4  11  25
   6   1   0   0   0  -1  -5 -16 -41
   8   2   1   1   1   1   2   7  23  64

Crossrefs

Rows are: A000009 (k=0), A087897 (k=1, without first term), A378972 (k=2).
For primes we have A095195 or A376682.
For partitions we have A175804.
First column is A293467 (up to sign).
For composites we have A377033.
For squarefree numbers we have A377038.
For nonsquarefree numbers we have A377046.
For prime powers we have A377051.
Position of first zero in each row is A377285.
Triangle's row-sums are A378970, absolute A378971.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A047966, A098859, A225486, A325244, A325282.
Cf. A008284, A116608, A325242, A225485 or A325280.

Programs

  • Mathematica
    nn=20;
t=Table[Take[Differences[PartitionsQ/@Range[0,2nn],k],nn],{k,0,nn}];
Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A377041 First term of the n-th differences of the squarefree numbers. Inverse zero-based binomial transform of A005117.

Original entry on oeis.org

1, 1, 0, 1, -3, 6, -8, 3, 22, -92, 252, -578, 1189, -2255, 3991, -6617, 10245, -14626, 18666, -19635, 12104, 13090, -69122, 171478, -332718, 552138, -798629, 982514, -901485, 116219, 2351842, -8715135, 23856206, -57926011, 130281064, -273804584, 535390333
Offset: 0

Views

Author

Gus Wiseman, Oct 18 2024

Keywords

Crossrefs

The version for primes is A007442, noncomposites A030016, composites A377036.
This is the first column of A377038.
For nonsquarefree numbers we have A377049.
For prime-powers we have A377054.
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.
A377042 gives first position of 0 in each row of A377038.
Cf. A053797, A053806, A061398, A072284, A076259, A120992, A376311, A376590, A376591, A377040, A377046.

Programs

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

Formula

The inverse zero-based binomial transform of a sequence (q(0), q(1), q(2), ...) is the sequence p given by:
p(j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) q(k)

A377036 First term of the n-th differences of the composite numbers. Inverse zero-based binomial transform of A002808.

Original entry on oeis.org

4, 2, 0, -1, 2, -2, 0, 4, -8, 8, 0, -16, 32, -32, -1, 78, -233, 687, -2363, 8160, -25670, 72352, -184451, 430937, -933087, 1888690, -3597221, 6479696, -11086920, 18096128, -28307626, 42644791, -62031001, 86466285, -110902034, 110907489, -52325, -483682930
Offset: 0

Views

Author

Gus Wiseman, Oct 18 2024

Keywords

Crossrefs

The version for prime instead of composite is A007442.
For noncomposite numbers we have A030016.
This is the first column (n=1) of A377033.
For row-sums we have A377034, absolute version A377035.
First zero positions are A377037, cf. A376678, A376855, A377042, A377050, A377055.
For squarefree instead of composite we have A377041, nonsquarefree A377049.
For prime-power instead of composite we have A377054.
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, A140119, A173390, A333214, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680.

Programs

  • Mathematica
    q=Select[Range[100],CompositeQ];
t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[1+k]],{k,0,j}],{j,0,Length[q]-1}]

Formula

The inverse zero-based binomial transform of a sequence (q(0), q(1), ..., q(m)) is the sequence p given by:
p(j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) q(k)

A378373 Number of composite numbers (A002808) between consecutive nonsquarefree numbers (A013929), exclusive.

Original entry on oeis.org

1, 0, 1, 2, 0, 0, 2, 0, 1, 0, 1, 3, 2, 1, 0, 1, 0, 0, 1, 0, 1, 2, 1, 0, 2, 2, 1, 0, 2, 0, 1, 3, 0, 1, 3, 0, 0, 0, 1, 2, 2, 2, 0, 2, 0, 2, 0, 0, 0, 2, 2, 0, 1, 3, 2, 0, 0, 0, 0, 2, 2, 1, 0, 2, 0, 1, 0, 1, 0, 2, 2, 3, 0, 1, 2, 0, 0, 3, 2, 0, 2, 3, 3, 2, 0, 1, 2
Offset: 1

Views

Author

Gus Wiseman, Dec 02 2024

Keywords

Comments

All terms are 0, 1, 2, or 3 (cf. A078147).
The inclusive version is a(n) + 2.
The nonsquarefree numbers begin: 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, ...

Examples

			The composite numbers counted by a(n) form the following set partition of A120944:
{6}, {}, {10}, {14,15}, {}, {}, {21,22}, {}, {26}, {}, {30}, {33,34,35}, {38,39}, ...

Crossrefs

For prime (instead of nonsquarefree) we have A046933.
For squarefree (instead of nonsquarefree) we have A076259(n)-1.
For prime power (instead of nonsquarefree) we have A093555.
For prime instead of composite we have A236575.
For nonprime prime power (instead of nonsquarefree) we have A378456.
For perfect power (instead of nonsquarefree) we have A378614, primes A080769.
A002808 lists the composite numbers.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A073247 lists squarefree numbers with nonsquarefree neighbors.
A120944 lists squarefree composite numbers.
A377432 counts perfect-powers between primes, zeros A377436.
A378369 gives distance to the next nonsquarefree number (A120327).
Cf. A065890, A067535, A067871, A080101, A081221, A151800, A243348, A366833, A377046, A378033, A378039, A378086.

Programs

  • Mathematica
    v=Select[Range[100],!SquareFreeQ[#]&];
Table[Length[Select[Range[v[[i]]+1,v[[i+1]]-1],CompositeQ]],{i,Length[v]-1}]

A378369 Distance between n and the least nonsquarefree number >= n.

Original entry on oeis.org

3, 2, 1, 0, 3, 2, 1, 0, 0, 2, 1, 0, 3, 2, 1, 0, 1, 0, 1, 0, 3, 2, 1, 0, 0, 1, 0, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 0, 2, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 3, 2, 1, 0, 2, 1, 0, 0, 3, 2, 1, 0, 3, 2, 1, 0, 2, 1, 0, 0, 3, 2, 1, 0, 0, 2, 1, 0, 3, 2, 1
Offset: 1

Views

Author

Gus Wiseman, Dec 01 2024

Keywords

Comments

All terms are 0, 1, 2, or 3 (cf. A078147).

Crossrefs

Adding n to each term a(n) gives A120327.
Positions of 0 are A013929.
Positions of 1 are A373415.
Positions of 2 are A378458.
Positions of 3 are A007675.
Sequences obtained by adding n to each term are placed in parentheses below.
The version for primes is A007920 (A007918).
The version for perfect powers is A074984 (A377468).
The version for squarefree numbers is A081221 (A067535).
The version for non-perfect powers is A378357 (A378358).
The version for prime powers is A378370 (A000015).
The version for non prime powers is A378371 (A378372).
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A120992 gives run-lengths of squarefree numbers increasing by one.
Cf. A073247, A151800, A236575, A243348, A375707, A377046, A378033, A378039, A378086, A378373.

Programs

  • Mathematica
    Table[NestWhile[#+1&,n,SquareFreeQ[#]&]-n,{n,100}]

A378458 Squarefree numbers k such that k + 1 is squarefree but k + 2 is not.

Original entry on oeis.org

2, 6, 10, 14, 22, 30, 34, 38, 42, 46, 58, 61, 66, 70, 73, 78, 82, 86, 94, 102, 106, 110, 114, 118, 122, 130, 133, 138, 142, 145, 154, 158, 166, 173, 178, 182, 186, 190, 194, 202, 205, 210, 214, 218, 222, 226, 230, 238, 246, 254, 258, 262, 266, 273, 277, 282
Offset: 1

Views

Author

Gus Wiseman, Dec 02 2024

Keywords

Comments

These are the positions of 2 in A378369 (difference between n and the next nonsquarefree number).
The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^2) - Product_{p prime} (1 - 3/p^2) = A065474 - A206256 = 0.19714711803343537224... . - Amiram Eldar, Dec 03 2024

Links

Crossrefs

Complement of A007675 within A007674.
The version for prime power instead of nonsquarefree is a subset of A006549.
Another variation is A073247.
The version for nonprime instead of squarefree is A179384.
Positions of 0 in A378369 are A013929.
Positions of 1 in A378369 are A373415.
Positions of 2 in A378369 are A378458 (this).
Positions of 3 in A378369 are A007675.
A000961 lists the powers of primes, differences A057820.
A120327 gives the least nonsquarefree number >= n.
A378373 counts composite numbers between nonsquarefree numbers.
Cf. A065474, A067535, A081221, A206256, A236575, A377046, A378033, A378039, A378086.

Programs

  • Mathematica
    Select[Range[100],NestWhile[#+1&,#,SquareFreeQ[#]&]==#+2&]
  • PARI
    list(lim) = my(q1 = 1, q2 = 1, q3); for(k = 3, lim, q3 = issquarefree(k); if(q1 && q2 &&!q3, print1(k-2, ", ")); q1 = q2; q2 = q3); \\ Amiram Eldar, Dec 03 2024

A379542 Second term of the n-th differences of the prime numbers.

Original entry on oeis.org

3, 2, 0, 2, -6, 14, -30, 62, -122, 220, -344, 412, -176, -944, 4112, -11414, 26254, -53724, 100710, -175034, 281660, -410896, 506846, -391550, -401486, 2962260, -9621128, 24977308, -57407998, 120867310, -236098336, 428880422, -719991244, 1096219280
Offset: 0

Views

Author

Gus Wiseman, Jan 12 2025

Keywords

Comments

Also the inverse zero-based binomial transform of the odd prime numbers.

Crossrefs

For all primes (not just odd) we have A007442.
Including 1 in the primes gives A030016.
Column n=2 of A095195.
The version for partitions is A320590 (first column A281425), see A175804, A053445.
For nonprime instead of prime we have A377036, see A377034-A377037.
Arrays of differences: A095195, A376682, A377033, A377038, A377046, A377051.
A000040 lists the primes, differences A001223, A036263.
A002808 lists the composite numbers, differences A073783, A073445.
A008578 lists the noncomposite numbers, differences A075526.
Cf. A064113, A065890, A084758, A140119, A173390, A258025, A258026, A293467, A333214, A333254, A377041.

Programs

  • Mathematica
    nn=40;Table[Differences[Prime[Range[nn+2]],n][[2]],{n,0,nn}]
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * prime(k+2)); \\ Michel Marcus, Jan 12 2025

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * prime(k+2).
Previous Showing 21-27 of 27 results.

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