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 1-7 of 7 results.

A278101 Triangle T(n,k) = A277648(n,k)^2 * A005117(k), read by rows.

Original entry on oeis.org

1, 4, 2, 3, 9, 8, 3, 5, 6, 7, 16, 8, 12, 5, 6, 7, 10, 11, 13, 14, 15, 25, 18, 12, 20, 24, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 36, 32, 27, 20, 24, 28, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 49, 32, 48, 45, 24, 28, 40, 44, 13, 14, 15, 17, 19, 21, 22
Offset: 1

Views

Author

Jason Kimberley, Nov 15 2016

Keywords

Comments

Other that the first (with length 1), row n has length A278100(n).
Equivalently, the surd sqrt(T(n,k)) = A277648(n,k) * sqrt(A005117(k)).

Examples

			The first five rows are:
1;
4,  2,  3;
9,  8,  3,  5,  6,  7;
16,  8, 12,  5,  6,  7, 10, 11, 13, 14, 15;
25, 18, 12, 20, 24,  7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23;

References

  • R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, submitted.

Links

Crossrefs

Cf. A278103.

Programs

  • Magma
    A277647:=func;
A278101_row:=funcA277647(n,j):j in[1..n^2]|IsSquarefree(j)]>;
&cat[A278101_row(n):n in[1..8]];
  • Mathematica
    DeleteCases[#, 0] & /@ Table[Boole[SquareFreeQ@ k] k Floor[n/Sqrt@ k]^2, {n, 7}, {k, n^2}] // Flatten (* Michael De Vlieger, Nov 24 2016 *)

A278104 Irregular triangle T(n,k) := A277648(n,k) for k = 1...A278102(n), read by rows.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 2, 5, 3, 2, 6, 4, 3, 2, 7, 4, 8, 5, 4, 3, 9, 6, 10, 7, 5, 11, 7, 12, 8, 6, 13, 9, 7, 5, 14, 9, 15, 10, 8, 6, 16, 11, 17, 12, 9, 18, 12, 19, 13, 10, 20, 14, 11, 8, 21, 14, 22, 15, 12, 9, 8, 23, 16, 13, 10, 9, 8, 24, 16, 13, 10, 9, 25, 17, 26, 18, 27, 19, 15, 28, 19
Offset: 1

Views

Author

Jason Kimberley, Nov 15 2016

Keywords

Comments

This triangle lists the "descending sequences across ranks" of Eggleton et al.

Examples

			The first 23 rows are:
1;
2,  1;
3,  2,  1;
4,  2;
5,  3,  2;
6,  4,  3,  2;
7,  4;
8,  5,  4,  3;
9,  6;
10,  7,  5;
11,  7;
12,  8,  6;
13,  9,  7,  5;
14,  9;
15, 10,  8,  6;
16, 11;
17, 12,  9;
18, 12;
19, 13, 10;
20, 14, 11,  8;
21, 14;
22, 15, 12,  9,  8;
23, 16, 13, 10,  9,  8;

References

  • R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, submitted.

Links

Programs

  • Magma
    A277647:=func;
A277648_row:=funcA277647(n,k):k in[1..n^2]|IsSquarefree(k)]>;
A278101_row:=funcA277647(n,k)^2*k:k in[1..n^2]|IsSquarefree(k)]>;
A278104_row:=funcA277648_row(n)[1..j]:j in[1..#row-1]|row[j]le row[j+1]}select dec else[1]) where row is A278101_row(n) >;
&cat[A278104_row(n):n in[1..23]];
  • Mathematica
    Map[Last, #, {2}] &@ Map[TakeWhile[FoldList[Function[s, Boole[s < 0] {First@ #2, Last@ #2}][First@ #2 - First@ #1] &, #], Total@ # > 0 &] &, #] &@ Map[DeleteCases[#, {0, 0}] &, Table[Boole[SquareFreeQ@ k] {k #^2, #} &@ Floor[n/Sqrt@ k], {n, 32}, {k, n^2}] ] // Flatten (* Michael De Vlieger, Nov 24 2016 *)

A277647 Triangle T(n,k) = floor(n/sqrt(k)) for 1 <= k <= n^2, read by rows.

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 4, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 4, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2
Offset: 1

Views

Author

Jason Kimberley, Nov 09 2016

Keywords

Examples

			The first five rows of the triangle are:
1;
2, 1, 1, 1;
3, 2, 1, 1, 1, 1, 1, 1, 1;
4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
5, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Links

Crossrefs

Cf. A010766, A277646, A277648.
The 1000th row is A033432.

Programs

  • Magma
    A277647:=func;
[A277647(n,k):k in[1..n^2],n in[1..7]];
  • Mathematica
    Table[Floor[n/Sqrt@ k], {n, 7}, {k, n^2}] // Flatten (* Michael De Vlieger, Nov 24 2016 *)
  • PARI
    row(n) = for(k=1, n^2, print1(floor(n/sqrt(k)), ", ")); print("")
trianglerows(n) = for(k=1, n, row(k))
/* Print initial five rows of triangle as follows: */
trianglerows(5) \\ Felix Fröhlich, Nov 12 2016

Formula

T(n,k) = A000196(A277646(n,k)).
T(n,k)sqrt(k) <= n < (T(n,k)+1)sqrt(k).

A277646 Triangle T(n,k) = floor(n^2/k) for 1 <= k <= n^2, read by rows.

Original entry on oeis.org

1, 4, 2, 1, 1, 9, 4, 3, 2, 1, 1, 1, 1, 1, 16, 8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 25, 12, 8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 36, 18, 12, 9, 7, 6, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 49, 24, 16, 12, 9, 8, 7, 6
Offset: 1

Views

Author

Jason Kimberley, Nov 09 2016

Keywords

Examples

			The first five rows of the triangle are:
1;
4, 2, 1, 1;
9, 4, 3, 2, 1, 1, 1, 1, 1;
16, 8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
25, 12, 8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Links

Crossrefs

Cf. Related triangles: A010766, A277647, A277648.
Rows of this triangle (with infinite trailing zeros):
T(1,k) = A000007(k-1),
T(2,k) = A033324(k),
T(3,k) = A033329(k),
T(4,k) = A033336(k),
T(5,k) = A033345(k),
T(6,k) = A033356(k),
T(7,k) = A033369(k),
T(8,k) = A033384(k),
T(9,k) = A033401(k),
T(10,k) = A033420(k),
T(100,k) = A033422(k),
T(10^3,k) = A033426(k),
T(10^4,k) = A033424(k).
Columns of this triangle:
T(n,1) = A000290(n),
T(n,2) = A007590(n),
T(n,3) = A000212(n),
T(n,4) = A002620(n),
T(n,5) = A118015(n),
T(n,6) = A056827(n),
T(n,7) = A056834(n),
T(n,8) = A130519(n+1),
T(n,9) = A056838(n),
T(n,10)= A056865(n),
T(n,12)= A174709(n+2).

Programs

  • Magma
    A277646:=func;
[A277646(n,k):k in[1..n^2],n in[1..7]];
  • Mathematica
    Table[Floor[n^2/k], {n, 7}, {k, n^2}] // Flatten (* Michael De Vlieger, Nov 24 2016 *)

Formula

T(n,k) = A010766(n^2,k).

A278100 Number of squarefree positive integers less than n^2.

Original entry on oeis.org

0, 3, 6, 11, 16, 23, 31, 39, 50, 61, 75, 89, 103, 120, 139, 157, 177, 199, 219, 243, 269, 297, 323, 351, 381, 412, 444, 477, 513, 547, 584, 624, 660, 703, 745, 789, 835, 882, 928, 977, 1025, 1073, 1124, 1174, 1230, 1285, 1342, 1400, 1460, 1523, 1582, 1645, 1708
Offset: 1

Views

Author

Jason Kimberley, Nov 12 2016

Keywords

Links

Crossrefs

Cf. A005117, A013928, A059956.
This is the row length sequence of A277648 and A278101.

Programs

  • Magma
    A278100:=func;
[A278100(n):n in[1..53]]; // in cubic time
  • Mathematica
    Table[Count[Range[n^2], k_ /; SquareFreeQ@ k], {n, 53}] (* Michael De Vlieger, Nov 24 2016 *)
Module[{nn=60,sf},sf=Accumulate[Table[If[SquareFreeQ[n],1,0],{n,0,nn^2}]];Table[sf[[k^2]],{k,nn}]] (* Harvey P. Dale, Nov 14 2020 *)
  • PARI
    a(n) = #select(x->issquarefree(x), vector(n^2-1, k, k)); \\ Michel Marcus, Nov 12 2016

Formula

a(n) = A013928(n^2).
a(n) ~ 6*n^2/Pi^2 + O(n). - Amiram Eldar, Mar 09 2021

A278113 Triangle T(n,k) = A278112(n,A000040(k)) for 1 <= k <= A278114(n), read by rows.

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 5, 4, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Jason Kimberley, Feb 09 2017

Keywords

Comments

This triangle consists of those columns of A278112 that have prime index.

Examples

			The first eight rows are:
  1;
  2, 1, 1, 1;
  3, 2, 1, 1, 1, 1, 1;
  4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1;
  5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  7, 5, 4, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Links

Crossrefs

Cf. A000040, A277648, A278112, A278114, A278115, A278118.

Programs

  • Magma
    A278112:=func;
A278113_row:=funcA278112(n,p):p in PrimesUpTo(2*n^2)]>;
&cat[A278113_row(n):n in[1..8]];
  • Mathematica
    Table[Floor[n Sqrt[2/Prime@ k]], {n, 8}, {k, PrimePi[2 n^2]}] // Flatten (* Michael De Vlieger, Feb 17 2017 *)

Formula

T(n,k) = floor(n*sqrt(2/prime(k))).
T(n,k) sqrt(A000040(k)) <= n sqrt(2) < (T(n,k)+1) sqrt(A000040(k)).

A278109 Irregular triangle read by rows: T(n,k) = floor(n/prime(k)^2) for 1 <= prime(k)^2 <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 3, 1, 3, 1, 4, 1, 4, 1, 4, 2, 4, 2, 5, 2, 5, 2, 5, 2, 5, 2, 6, 2, 6, 2, 1, 6, 2, 1, 6, 3, 1, 7, 3, 1, 7, 3, 1, 7, 3, 1, 7, 3, 1, 8, 3, 1, 8, 3, 1, 8, 3, 1, 8, 3, 1, 9, 4, 1, 9, 4, 1, 9, 4, 1, 9, 4, 1, 10, 4, 1, 10, 4, 1, 10, 4, 1, 10, 4, 1, 11, 4, 1, 11, 5, 1
Offset: 4

Views

Author

Jason Kimberley, Feb 02 2017

Keywords

Comments

This triangle consists of those columns of A278108 with prime index.
The row length sequence is A056811.

Examples

			The initial rows (for n = 4..27) are:
1;
1;
1;
1;
2;
2, 1;
2, 1;
2, 1;
3, 1;
3, 1;
3, 1;
3, 1;
4, 1;
4, 1;
4, 2;
4, 2;
5, 2;
5, 2;
5, 2;
5, 2;
6, 2;
6, 2, 1;
6, 2, 1;
6, 3, 1;

Links

Crossrefs

Cf. A056811, A277648, A278108, A278110.

Programs

  • Magma
    [n div p^2:p in PrimesUpTo(Isqrt(n)),n in[1..45]];

Formula

T(n,k) = A278108(n,A000040(k)).
Showing 1-7 of 7 results.

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