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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A376591 Inflection and undulation points in the sequence of squarefree numbers (A005117).

Original entry on oeis.org

1, 4, 9, 11, 12, 14, 16, 18, 21, 24, 27, 32, 33, 35, 40, 43, 48, 53, 55, 56, 58, 62, 65, 68, 71, 79, 84, 87, 96, 98, 99, 101, 103, 107, 110, 113, 118, 120, 121, 123, 128, 131, 134, 137, 142, 144, 145, 147, 152, 153, 155, 158, 163, 165, 166, 172, 175, 179, 184
Offset: 1

Views

Author

Gus Wiseman, Oct 04 2024

Keywords

Comments

These are points at which the second differences (A376590) are zero.

Examples

			The squarefree numbers (A005117) are:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, ...
with first differences (A076259):
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, ...
with first differences (A376590):
  0, 1, -1, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 2, 0, -2, 0, 1, -1, 0, 1, -1, 0, 1, ...
with zeros at (A376591):
 1, 4, 9, 11, 12, 14, 16, 18, 21, 24, 27, 32, 33, 35, 40, 43, 48, 53, 55, 56, 58, ...

Links

Crossrefs

The first differences were A076259, see also A375927, A376305, A376306, A376307, A376311.
These are the zeros of A376590.
The complement is A376592.
A000040 lists the prime numbers, differences A001223.
A005117 lists squarefree numbers, complement A013929 (differences A078147).
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
For inflections and undulations: A064113 (prime), A376602 (composite), A376588 (non-perfect-power), A376594 (nonsquarefree), A376597 (prime-power), A376600 (non-prime-power).
For squarefree numbers: A076259 (first differences), A376590 (second differences), A376592 (nonzero curvature).
Cf. A007674, A036263, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A333254, A376593, A376655.

Programs

  • Mathematica
    Join@@Position[Differences[Select[Range[100],SquareFreeQ],2],0]

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}]

A376604 Second differences of the Kolakoski sequence (A000002). First differences of A054354.

Original entry on oeis.org

-1, -1, 1, 1, -2, 2, -1, -1, 2, -1, -1, 1, 1, -2, 1, 1, -1, -1, 2, -2, 1, 1, -2, 2, -1, -1, 1, 1, -2, 1, 1, -2, 2, -1, -1, 2, -1, -1, 1, 1, -2, 2, -1, -1, 2, -2, 1, 1, -2, 1, 1, -1, -1, 2, -1, -1, 1, 1, -2, 2, -1, -1, 2, -1, -1, 1, 1, -2, 1, 1, -2, 2, -1, -1
Offset: 1

Views

Author

Gus Wiseman, Oct 02 2024

Keywords

Comments

Since A000002 has no runs of length 3, this sequence contains no zeros.
The densities appear to approach (1/3, 1/3, 1/6, 1/6).

Examples

			The Kolakoski sequence (A000002) is:
  1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, ...
with first differences (A054354):
  1, 0, -1, 0, 1, -1, 1, 0, -1, 1, 0, -1, 0, 1, -1, 0, 1, 0, -1, 1, -1, 0, 1, -1, ...
with first differences (A376604):
  -1, -1, 1, 1, -2, 2, -1, -1, 2, -1, -1, 1, 1, -2, 1, 1, -1, -1, 2, -2, 1, 1, -2, ...

Crossrefs

A001462 is Golomb's sequence.
A078649 appears to be zeros of the first and third differences.
A288605 gives positions of first appearances of each balance.
A306323 gives a 'broken' version.
A333254 lists run-lengths of differences between consecutive primes.
For the Kolakoski sequence (A000002):
- Restrictions: A074264, A100428, A100429, A156263, A156264.
- Patterns: A013947, A013948, A022292, A074262, A074263, A156242, A156243.
- Statistics: A054353, A074286, A088568, A156077, A156253.
- Transformations: A054354, A156728, A332273, A332875, A333229, A376604.
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power), A376599 (non-prime-power).

Programs

  • Mathematica
    kolagrow[q_]:=If[Length[q]<2,Take[{1,2},Length[q]+1],Append[q,Switch[{q[[Length[Split[q]]]],q[[-2]],Last[q]},{1,1,2},1,{1,2,1},2,{2,1,1},2,{2,1,2},2,{2,2,1},1,{2,2,2},1]]]
kol[n_]:=Nest[kolagrow,{1},n-1];
Differences[kol[100],2]

A376592 Points of nonzero curvature in the sequence of squarefree numbers (A005117).

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 13, 15, 17, 19, 20, 22, 23, 25, 26, 28, 29, 30, 31, 34, 36, 37, 38, 39, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 54, 57, 59, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 85, 86, 88, 89, 90, 91, 92, 93, 94, 95
Offset: 1

Views

Author

Gus Wiseman, Oct 04 2024

Keywords

Comments

These are points at which the second differences (A376590) are nonzero.

Examples

			The squarefree numbers (A005117) are:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, ...
with first differences (A076259):
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, ...
with first differences (A376590):
  0, 1, -1, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 2, 0, -2, 0, 1, -1, 0, 1, -1, 0, 1, ...
with nonzeros at (A376591):
  2, 3, 5, 6, 7, 8, 10, 13, 15, 17, 19, 20, 22, 23, 25, 26, 28, 29, 30, 31, 34, 36, ...

Links

Crossrefs

The first differences were A076259, see also A375927, A376305, A376306, A376307, A376311.
These are the nonzeros of A376590.
The complement is A376591.
A000040 lists the prime numbers, differences A001223.
A005117 lists squarefree numbers, complement A013929 (differences A078147).
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
For points of nonzero curvature: A333214 (prime), A376603 (composite), A376589 (non-perfect-power), A376595 (nonsquarefree), A376598 (prime-power), A376601 (non-prime-power).
For squarefree numbers: A076259 (first differences), A376590 (second differences), A376591 (inflection and undulation points).
Cf. A007674, A036263, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A333254, A373198, A376655.

Programs

  • Mathematica
    Join@@Position[Sign[Differences[Select[Range[100], SquareFreeQ],2]],1|-1]

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}]

A377042 Position of first zero in the n-th differences of the squarefree numbers (A005117), or 0 if it does not appear.

Original entry on oeis.org

0, 0, 1, 11, 8, 57, 14, 11, 13, 1019, 44, 1250, 43, 2721, 42, 249522, 2840, 1989839, 2839, 3373774, 4933, 142715511, 42793, 435650856, 5266, 30119361, 104063, 454172978707, 100285, 434562125244, 2755089, 2409925829164, 2485612
Offset: 0

Views

Author

Gus Wiseman, Oct 18 2024

Keywords

Comments

a(n) for n even appear to be smaller than a(n) for n odd. - Chai Wah Wu, Oct 19 2024
a(33) > 10^13, unless it is 0. - Lucas A. Brown, Nov 15 2024

Examples

			The fourth differences begin: -3, 3, 1, -6, 7, -5, 3, 0, -2, ... so a(4) = 8

Links

Crossrefs

The version for primes is A376678, noncomposites A376855, composites A377037.
This is the first position of 0 in each row of A377038.
For nonsquarefree numbers we have A377050.
For prime-powers we have A377055.
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.
A377039 gives antidiagonal-sums of A377038, absolute version A377040.
A377041 gives first column of A377038, for primes A007442 or A030016.
Cf. A007674, A053797, A053806, A061398, A072284, A076259, A112925, A120992, A376311, A376590, A376591, A377046.

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(15)-a(20) from Chai Wah Wu, Oct 19 2024
a(21)-a(32) from Lucas A. Brown, Nov 15 2024

A376653 Sorted positions of first appearances in the second differences of consecutive prime-powers inclusive (A000961).

Original entry on oeis.org

1, 4, 5, 10, 12, 18, 25, 45, 47, 48, 60, 68, 69, 71, 80, 118, 121, 178, 179, 199, 206, 207, 216, 244, 245, 304, 325, 327, 402, 466, 484, 605, 801, 880, 939, 1033, 1055, 1077, 1234, 1281, 1721, 1890, 1891, 1906, 1940, 1960, 1962, 2257, 2290, 2410, 2880, 3150
Offset: 1

Views

Author

Gus Wiseman, Oct 06 2024

Keywords

Examples

			The prime-powers inclusive (A000961) are:
  1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, ...
with first differences (A057820):
  1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, ...
with first differences (A376596):
  0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, ...
with first appearances (A376653):
  1, 4, 5, 10, 12, 18, 25, 45, 47, 48, 60, 68, 69, 71, 80, 118, 121, 178, 179, 199, ...

Crossrefs

For first differences we had A057820, sorted firsts A376340(n)+1 (except first term).
These are the sorted positions of first appearances in A376596.
The exclusive version is a(n) - 1 = A376654(n), except first term.
For squarefree instead of prime-power we have A376655.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
A064113 lists positions of adjacent equal prime gaps.
For prime-powers inclusive: A057820 (first differences), A376597 (inflections and undulations), A376598 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).
Cf. A053707, A069623, A093555, A174965, A251092, A333254, A361102.

Programs

  • Mathematica
    q=Differences[Select[Range[100],#==1||PrimePowerQ[#]&],2];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A378972 Second differences of the strict partition numbers A000009.

Original entry on oeis.org

0, 1, -1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 2, 0, 2, 2, 1, 2, 3, 2, 3, 4, 3, 4, 6, 4, 6, 8, 6, 9, 10, 9, 12, 14, 13, 16, 19, 18, 22, 26, 24, 30, 34, 34, 40, 45, 46, 53, 60, 62, 70, 79, 82, 93, 104, 108, 122, 136, 142, 160, 176, 186, 208, 228, 243, 268
Offset: 0

Views

Author

Gus Wiseman, Dec 14 2024

Keywords

Examples

			The strict partition numbers begin (A000009):
  1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, ...
with differences (A087897 without first term):
  0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 8, 8, 10, 12, ...
with differences (a(n)):
  0, 1, -1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 2, 0, 2, 2, 1, 2, ...

Crossrefs

For primes we have A036263.
The version for partitions is A053445.
For composites we have A073445.
For squarefree numbers we have A376590.
For nonsquarefree numbers we have A376593.
For powers of primes (inclusive) we have A376596.
For non powers of primes (inclusive) we have A376599.
Second row of A378622. See also:
- A293467 gives first column (up to sign).
- A377285 gives position of first zero in each row.
- A378970 gives row-sums.
- A378971 gives absolute value row-sums.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A047966, A098859, A225486, A325244, A325282.
Cf. A008284, A116608, A225485, A325242, A325268.

Programs

  • Mathematica
    Differences[Table[PartitionsQ[n],{n,0,100}],2]

A376560 Points of upward concavity in the sequence of perfect-powers (A001597). Positives of A376559.

Original entry on oeis.org

1, 3, 4, 6, 8, 9, 10, 11, 12, 15, 16, 17, 20, 22, 23, 26, 27, 28, 31, 32, 33, 34, 35, 36, 37, 38, 41, 42, 43, 44, 46, 47, 48, 49, 50, 53, 54, 55, 57, 58, 60, 61, 62, 63, 64, 65, 67, 68, 69, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91
Offset: 1

Views

Author

Gus Wiseman, Sep 30 2024

Keywords

Comments

These are points at which the second differences are positive.
Perfect-powers (A001597) are numbers with a proper integer root.
Note that, for some sources, upward concavity is negative curvature.

Examples

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, 33, ...
with first differences (A376559):
  1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, 2, ...
with positive positions (A376560):
  1, 3, 4, 6, 8, 9, 10, 11, 12, 15, 16, 17, 20, 22, 23, 26, 27, 28, 31, 32, 33, 34, ...

Links

Crossrefs

The version for A000002 is A022297, complement A025505. See also A054354, A376604.
For first differences we have A053289, union A023055, firsts A376268, A376519.
For primes instead of perfect-powers we have A258025.
These are positions of positive terms in A376559.
For downward concavity we have A376561 (probably the complement).
A001597 lists the perfect-powers.
A064113 lists positions of adjacent equal prime gaps.
A333254 gives run-lengths of differences between consecutive primes.
Second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power), A376599 (non-prime-power).
Cf. A006549, A025475, A030173, A045542, A052410, A053707, A054819, A069623, A174965, A216765, A251092, A376308.

Programs

  • Maple
    N:= 10^6: # to use perfect powers <= N
S:= {1,seq(seq(i^j,j=2..floor(log[i](N))),i=2..isqrt(N))}:
L:= sort(convert(S,list)):
DL:= L[2..-1]-L[1..-2]:
D2L:= DL[2..-1]-DL[1..-2]:
select(i -> D2L[i]>0, [$1..nops(D2L)]); # Robert Israel, Dec 01 2024
  • Mathematica
    perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Join@@Position[Sign[Differences[Select[Range[1000],perpowQ],2]],1]

A376654 Sorted positions of first appearances in the second differences of consecutive prime-powers exclusive (A246655).

Original entry on oeis.org

3, 4, 9, 11, 17, 24, 44, 46, 47, 59, 67, 68, 70, 79, 117, 120, 177, 178, 198, 205, 206, 215, 243, 244, 303, 324, 326, 401, 465, 483, 604, 800, 879, 938, 1032, 1054, 1076, 1233, 1280, 1720, 1889, 1890, 1905, 1939, 1959, 1961, 2256, 2289, 2409, 2879, 3149
Offset: 1

Views

Author

Gus Wiseman, Oct 06 2024

Keywords

Examples

			The prime-powers exclusive (A246655) are:
  2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, ...
with first differences (A057820 except first term) :
  1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, 3, ...
with first differences (A376596 except first term):
  0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, -4, ...
with first appearances (A376654):
  1, 3, 4, 9, 11, 17, 24, 44, 46, 47, 59, 67, 68, 70, 79, 117, 120, 177, 178, 198, ...

Crossrefs

For first differences we have A376340.
These are the sorted positions of first appearances in A376596 except first term.
The inclusive version is a(n) + 1 = A376653(n), except first term.
For squarefree instead of prime-power we have A376655.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
For prime-powers inclusive: A057820 (first differences), A376597 (inflections and undulations), A376598 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).
Cf. A025475, A053707, A057820, A064113, A069623, A093555, A174965, A333254, A336416, A361102.

Programs

  • Mathematica
    q=Differences[Select[Range[1000],PrimePowerQ[#]&],2];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]
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