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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A377052 Antidiagonal-sums of the array A377051(n,k) = n-th term of k-th differences of powers of primes.

Original entry on oeis.org

1, 3, 4, 5, 6, 13, -6, 45, -50, 113, -98, 73, 274, -1159, 3563, -8707, 19024, -36977, 64582, -98401, 121436, -81961, -147383, 860871, -2709964, 7110655, -17077217, 38873213, -85085216, 179965720, -367884935, 725051361, -1372311916, 2481473639, -4257624155
Offset: 0

Views

Author

Gus Wiseman, Oct 22 2024

Keywords

Comments

These are the row-sums of the triangle-version of A377051.

Examples

			The sixth antidiagonal of A377051 is (8, 1, -1, -2, -3, -4, -5), so a(6) = -6.

Crossrefs

The version for primes is A140119, noncomposites A376683, composites A377034.
For squarefree numbers we have A377039, nonsquarefree A377047.
These are the antidiagonal-sums of A377051.
The unsigned version is A377053.
For leaders we have A377054, for primes A007442 or A030016.
For first zero-positions we have A377055.
A version for partitions is A377056, cf. A175804, A053445, A281425, A320590.
A000040 lists the primes, differences A001223, seconds A036263.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
Cf. A000961, A025475, A053707, A057820, A093555, A174965, A246655, A361102, A376340, A376596, A376598.

Programs

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

A379311 Number of prime indices of n that are 1 or prime.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 0, 3, 2, 2, 1, 3, 0, 1, 2, 4, 1, 3, 0, 3, 1, 2, 0, 4, 2, 1, 3, 2, 0, 3, 1, 5, 2, 2, 1, 4, 0, 1, 1, 4, 1, 2, 0, 3, 3, 1, 0, 5, 0, 3, 2, 2, 0, 4, 2, 3, 1, 1, 1, 4, 0, 2, 2, 6, 1, 3, 1, 3, 1, 2, 0, 5, 0, 1, 3, 2, 1, 2, 0, 5, 4, 2, 1, 3, 2, 1, 1
Offset: 1

Views

Author

Gus Wiseman, Dec 27 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 39 are {2,6}, so a(39) = 1.
The prime indices of 70 are {1,3,4}, so a(70) = 2.
The prime indices of 98 are {1,4,4}, so a(98) = 1.
The prime indices of 294 are {1,2,4,4}, a(294) = 2.
The prime indices of 1911 are {2,4,4,6}, so a(1911) = 1.
The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 2.

Crossrefs

Positions of first appearances are A000079.
These "old" primes are listed by A008578.
Positions of zero are A320629, counted by A023895 (strict A204389).
Positions of one are A379312, counted by A379314 (strict A379315).
Positions of nonzero terms are A379313.
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.
A080339 is the characteristic function for the old prime numbers.
A376682 gives k-th differences of old prime numbers, see A030016, A075526, A173390, A376683, A376855.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
Cf. A034891, A036234, A036497, A038550, A087436, A302540, A379300, A379301.

Programs

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

Formula

Totally additive with a(prime(k)) = A080339(k).

A377056 Antidiagonal-sums of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).

Original entry on oeis.org

1, 1, 4, 3, 11, 2, 36, -27, 142, -207, 595, -1066, 2497, -4878, 10726, -22189, 48383, -103318, 224296, -480761, 1030299, -2186942, 4626313, -9740648, 20492711, -43109372, 90843475, -191769296, 405528200, -858373221, 1817311451, -3845483855, 8129033837
Offset: 0

Views

Author

Gus Wiseman, Dec 12 2024

Keywords

Examples

			Antidiagonal i + j = 3 of A175804 is (3, 1, 0, -1), so a(3) = 3.

Crossrefs

For primes we have A140119 or A376683, unsigned A376681 or A376684.
These are the antidiagonal-sums of A175804.
First column of the same array is A281425.
For composites we have A377034, unsigned A377035.
For squarefree numbers we have A377039, unsigned A377040.
For nonsquarefree numbers we have A377049, unsigned A377048.
For prime powers we have A377052, unsigned A377053.
The unsigned version is A378621.
The version for strict partitions is A378970 (row-sums of A378622), unsigned A378971.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A075526, A116608, A225486, A293467, A325242, A325245, A325268, A377285.

Programs

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

A378970 Antidiagonal-sums of the array A378622(n,k) = n-th term of k-th differences of strict partition numbers (A000009).

Original entry on oeis.org

1, 1, 1, 5, -4, 18, -20, 47, -56, 110, -153, 309, -532, 1045, -1768, 2855, -3620, 2928, 2927, -20371, 62261, -148774, 314112, -613835, 1155936, -2175658, 4244218, -8753316, 19006746, -42471491, 95234915, -210395017, 453414314, -949507878, 1931940045
Offset: 0

Views

Author

Gus Wiseman, Dec 14 2024

Keywords

Examples

			Antidiagonal 4 of A378622 is (2, 0, -1, -2, -3), so a(4) = -4.

Crossrefs

For primes we have A140119 or A376683, absolute value A376681 or A376684.
For composites we have A377034, absolute value A377035.
For squarefree numbers we have A377039, absolute value A377040.
For nonsquarefree numbers we have A377047, absolute value A377048.
For prime powers we have A377052, absolute value A377053.
For partition numbers we have A377056, absolute value A378621.
Row-sums of the triangular form of A378622. See also:
- A175804 is the version for partitions.
- A293467 gives the first column (up to sign).
- A377285 gives position of first zero in each row.
The unsigned version is A378971.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A098859, A116608, A225486, A325242, A325244.

Programs

  • Mathematica
    nn=30;
t=Table[Take[Differences[PartitionsQ/@Range[0,2nn],k],nn],{k,0,nn}];
Total/@Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A378971 Antidiagonal-sums of absolute value of the array A378622(n,k) = n-th term of k-th differences of strict partition numbers (A000009).

Original entry on oeis.org

1, 1, 1, 5, 8, 18, 30, 47, 70, 110, 177, 309, 574, 1063, 1892, 3107, 4598, 6166, 8737, 20603, 62457, 149132, 314116, 614093, 1155968, 2176048, 4244322, 8753864, 19006756, 42472117, 95235017, 210396059, 453414950, 949510166, 1931941261, 3826650257, 7400745917
Offset: 0

Views

Author

Gus Wiseman, Dec 14 2024

Keywords

Examples

			Antidiagonal 4 of A378622 is (2, 0, -1, -2, -3), so a(4) = 8.

Crossrefs

For primes we have A376681 or A376684, signed version A140119 or A376683.
For composites we have A377035, signed version A377034.
For squarefree numbers we have A377040, signed version A377039.
For nonsquarefree numbers we have A377048, signed version A377049.
For prime powers we have A377053, signed version A377052.
For partition numbers we have A378621, signed version A377056.
Row-sums of the triangular form of A378622. See also:
- A175804 is the version for partitions.
- A293467 gives the first column (up to sign).
- A377285 gives position of first zero in each row.
The signed version is A378970.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A098859, A116608, A225486, A325242, A325244, A325245, A325268.

Programs

  • Mathematica
    nn=30;
t=Table[Take[Differences[PartitionsQ/@Range[0,2nn],k],nn],{k,0,nn}];
Total/@Abs/@Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A378621 Antidiagonal-sums of absolute value of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).

Original entry on oeis.org

1, 1, 4, 5, 11, 16, 36, 65, 142, 285, 595, 1210, 2497, 5134, 10726, 22637, 48383, 104066, 224296, 481985, 1030299, 2188912, 4626313, 9743750, 20492711, 43114180, 90843475, 191776658, 405528200, 858384333, 1817311451, 3845500427, 8129033837, 17162815092
Offset: 0

Views

Author

Gus Wiseman, Dec 14 2024

Keywords

Examples

			Antidiagonal i + j = 3 of A175804 is (3, 1, 0, -1), so a(3) = 5.

Crossrefs

These are the antidiagonal-sums of the absolute value of A175804.
First column of the same array is A281425.
For primes we have A376681 or A376684, signed A140119 or A376683.
For composites we have A377035, signed A377034.
For squarefree numbers we have A377040, signed A377039.
For nonsquarefree numbers we have A377048, signed A377049.
For prime powers we have A377053, signed A377052.
The signed version is A377056.
The corresponding array for strict partitions is A378622, see A293467, A377285, A378971, A378970.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A075526, A098859, A116608, A225486, A325242, A325244, A325268.

Programs

  • Mathematica
    nn=30;
q=Table[PartitionsP[n],{n,0,nn}];
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/@Abs/@Table[t[[j,i-j+1]],{i,nn/2},{j,i}]
Previous Showing 11-16 of 16 results.

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