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 11 results. Next

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).

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

Views

Author

Gus Wiseman, Oct 19 2024

Keywords

Comments

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

Examples

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

Links

Crossrefs

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.

Programs

  • Mathematica
    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]]]}]

Extensions

a(17)-a(28) from Lucas A. Brown, Oct 25 2024

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

Views

Author

Gus Wiseman, Oct 17 2024

Keywords

Examples

			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.

Links

Crossrefs

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.

Programs

  • Mathematica
    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]]]}]

Extensions

Offset 2 from Michel Marcus, Oct 18 2024
a(17)-a(54) from Alois P. Heinz, Oct 18 2024

A377040 Antidiagonal-sums of absolute value of the array A377038(n,k) = n-th term of k-th differences of squarefree numbers (A005117).

Original entry on oeis.org

1, 3, 4, 9, 13, 18, 28, 39, 106, 267, 595, 1212, 2286, 4041, 6720, 10497, 15387, 20914, 25894, 29377, 37980, 70785, 175737, 343806, 579751, 861934, 1162080, 1431880, 1688435, 2589533, 8731932, 23911101, 58109574, 130912573, 276067892, 543833014, 992784443
Offset: 0

Views

Author

Gus Wiseman, Oct 18 2024

Keywords

Examples

			The fourth antidiagonal of A377038 is (6, 1, -1, -2, -3), so a(4) = 13.

Crossrefs

The version for primes is A376681, noncomposites A376684, composites A377035.
These are the antidiagonal-sums of the absolute value of A377038.
The non-absolute version is A377039.
For nonsquarefree numbers we have A377048, non-absolute A377047.
For prime-powers we have A377053, non-absolute A377052.
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.
A377041 gives first column of A377038, for primes A007442 or A030016.
A377042 gives first position of 0 in each row of A377038.
Cf. A007674, A053797, A053806, A061398, A072284, A075526, A076259, A120992, A140119, A376311, A376590, A376591, A377046.

Programs

  • Mathematica
    nn=20;
t=Table[Take[Differences[NestList[NestWhile[#+1&,#+1,!SquareFreeQ[#]&]&,1,2*nn],k],nn],{k,0,nn}];
Total/@Table[Abs[t[[j,i-j+1]]],{i,nn},{j,i}]

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

Views

Author

Gus Wiseman, Oct 14 2024

Keywords

Comments

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

Examples

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

Links

Crossrefs

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.

Programs

  • Mathematica
    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]]]}]

Formula

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

Extensions

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

Views

Author

Gus Wiseman, Oct 15 2024

Keywords

Examples

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

Links

Crossrefs

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.

Programs

  • Mathematica
    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]]]}]

Extensions

a(16)-a(21) from Alois P. Heinz, Oct 18 2024
a(22)-a(35) from Lucas A. Brown, Nov 03 2024

A377039 Antidiagonal-sums of the array A377038(n,k) = n-th term of k-th differences of squarefree numbers (A005117).

Original entry on oeis.org

1, 3, 4, 9, 1, 18, 8, -9, 106, -237, 595, -1170, 2276, -3969, 6640, -10219, 14655, -18636, 19666, -12071, -13056, 69157, -171441, 332756, -552099, 798670, -982472, 901528, -116173, -2351795, 8715186, -23856153, 57926066, -130281007, 273804642, -535390274
Offset: 0

Views

Author

Gus Wiseman, Oct 18 2024

Keywords

Comments

These are row-sums of the triangle-version of A377038.

Examples

			The fourth antidiagonal of A377038 is (6,1,-1,-2,-3), so a(4) = 1.

Crossrefs

The version for primes is A140119, noncomposites A376683, composites A377034.
These are the antidiagonal-sums of A377038.
The absolute version is A377040.
For nonsquarefree numbers we have A377047.
For prime-powers we have A377052.
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.
A377041 gives first column of A377038, for primes A007442 or A030016.
A377042 gives first position of 0 in each row of A377038.
Cf. A007674, A053797, A053806, A061398, A072284, A075526, A076259, A120992, A376311, A376590, A376591, A377046.

Programs

  • Mathematica
    nn=20;
t=Table[Take[Differences[NestList[NestWhile[#+1&,#+1,!SquareFreeQ[#]&]&,1,2*nn],k],nn],{k,0,nn}];
Total/@Table[t[[j,i-j+1]],{i,nn},{j,i}]

A377055 Position of first appearance of zero in the n-th differences of the prime-powers (A246655), or 0 if it does not appear.

Original entry on oeis.org

0, 0, 1, 1, 4, 48, 61, 83, 29, 57, 290, 121, 7115, 14207, 68320, 14652, 149979, 122704, 481540, 980376, 632441, 29973, 25343678, 50577935, 7512418, 210836403, 67253056, 224083553, 910629561, 931524323, 452509699, 2880227533, 396690327, 57954538325, 77572935454, 35395016473
Offset: 0

Views

Author

Gus Wiseman, Oct 22 2024

Keywords

Examples

			The fourth differences of A246655 begin: 1, -3, 3, 0, -2, 2, ... so a(4) = 4.

Links

Crossrefs

The version for primes is A376678, noncomposites A376855, composites A377037.
For squarefree numbers we have A377042, nonsquarefree A377050.
These are the positions of first zeros in each row of A377051.
For antidiagonal-sums we have A377052, absolute A377053.
For leaders we have A377054, for primes A007442 or A030016.
A000040 lists the primes, differences A001223, seconds A036263.
A000961 lists the powers of primes, differences A057820.
A008578 lists the noncomposites, differences A075526.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
A246655 lists the prime-powers, differences A057820 (except first term).
Cf. A025475, A053707, A093555, A174965, A175804, A361102, A376340, A376596, A377056.

Programs

  • Mathematica
    nn=10000;
u=Table[Differences[Select[Range[nn],PrimePowerQ],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]]]}]

Extensions

a(12)-a(27) from Pontus von Brömssen, Oct 22 2024
a(28)-a(30) from Chai Wah Wu, Oct 23 2024
a(31)-a(35) from Lucas A. Brown, Nov 03 2024

A377285 Position of first 0 in the n-th differences of the strict partition numbers A000009, or 0 if 0 does not appear.

Original entry on oeis.org

0, 1, 1, 5, 5, 8, 20, 7, 22
Offset: 0

Views

Author

Gus Wiseman, Dec 12 2024

Keywords

Comments

Open problem: Do the 9th differences of the strict integer partition numbers contain a zero? If so, we must have a(9) > 10^5.
a(12) = 47. Conjecture: a(n) = 0 for n > 12. - Chai Wah Wu, Dec 15 2024

Examples

			The 7th differences of A000009 are: 25, -16, 7, -6, 10, -9, 0, 10, ... so a(7) = 7.

Crossrefs

For primes we have A376678.
For composites we have A377037.
For squarefree numbers we have A377042.
For nonsquarefree numbers we have A377050.
For prime-powers we have A377055.
Position of first zero in each row of A378622. See also:
- A175804 is the version for partitions.
- A293467 gives first column (up to sign).
- A378970 gives row-sums.
- A378971 gives row-sums of absolute value.
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, A325268, A225485 or A325280.

Programs

  • Mathematica
    Table[Position[Differences[PartitionsQ/@Range[0,100],k],0][[1,1]],{k,1,8}]
  • PARI
    a(n, nn=100) = my(q='q+O('q^nn), v=Vec(eta(q^2)/eta(q))); for (i=1, n, my(w=vector(#v-1, k, v[k+1]-v[k])); v = w;); my(vz=select(x->x==0, v, 1)); if (#vz, vz[1]); \\ Michel Marcus, Dec 15 2024

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)
Showing 1-10 of 11 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.