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.

Previous Showing 11-19 of 19 results.

A344085 Triangle of squarefree numbers first grouped by greatest prime factor, then sorted by omega, then in increasing order, read by rows.

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 35, 42, 70, 105, 210, 11, 22, 33, 55, 77, 66, 110, 154, 165, 231, 385, 330, 462, 770, 1155, 2310, 13, 26, 39, 65, 91, 143, 78, 130, 182, 195, 273, 286, 429, 455, 715, 1001, 390, 546, 858, 910, 1365, 1430, 2002, 2145, 3003, 5005, 2730, 4290, 6006, 10010, 15015, 30030
Offset: 1

Views

Author

Gus Wiseman, May 11 2021

Keywords

Comments

Differs from A339195 in having 77 before 66.

Examples

			Triangle begins:
   1
   2
   3   6
   5  10  15  30
   7  14  21  35  42  70 105 210

Crossrefs

Row lengths are A000079.
Grouping by greatest prime factor only gives A339195.
Row sums are 1 and A339360.
Cf. A001221, A005117, A005183, A014466, A019565, A187769, A209862.
Partition/composition orderings: A026791, A026792, A026793, A036036, A036037, A048793, A066099, A080577, A112798, A118457, A124734, A162247, A193073, A211992, A228100, A228531, A246688, A272020, A299755, A296774, A304038, A319247, A329631, A334301, A334302, A334439, A334442, A335122, A344086, A344087, A344088, A344089.
Partition/composition applications: A001793, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124.

Programs

  • Mathematica
    nn=4;
GatherBy[SortBy[Select[Range[Times@@Prime/@Range[nn]],SquareFreeQ[#]&&PrimePi[FactorInteger[#][[-1,1]]]<=nn&],PrimeOmega],FactorInteger[#][[-1,1]]&]

A265146 Triangle T(n,k) in which n-th row lists the parts i_1=1, 1<=k<=A001222(n).

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 3, 4, 1, 2, 3, 2, 3, 1, 4, 5, 1, 2, 4, 6, 1, 5, 2, 4, 1, 2, 3, 4, 7, 1, 3, 4, 8, 1, 2, 5, 2, 5, 1, 6, 9, 1, 2, 3, 5, 3, 4, 1, 7, 2, 3, 4, 1, 2, 6, 10, 1, 3, 5, 11, 1, 2, 3, 4, 5, 2, 6, 1, 8, 3, 5, 1, 2, 4, 5, 12, 1, 9, 2, 7, 1, 2, 3, 6, 13, 1
Offset: 1

Views

Author

Alois P. Heinz, Dec 02 2015

Keywords

Comments

A strict partition is a partition into distinct parts.
Row n=1 contains the parts of the empty partition, so it is empty.

Examples

			n = 12 = 2*2*3 = prime(1)*prime(1)*prime(2) encodes strict partition [1,2,4].
Triangle T(n,k) begins:
01 :  ;
02 :  1;
03 :  2;
04 :  1, 2;
05 :  3;
06 :  1, 3;
07 :  4;
08 :  1, 2, 3;
09 :  2, 3;
10 :  1, 4;
11 :  5;
12 :  1, 2, 4;
13 :  6;
14 :  1, 5;
15 :  2, 4;
16 :  1, 2, 3, 4;

Links

Crossrefs

Column k=1 gives A055396 (for n>1).
Last terms of rows give A252464 (for n>1).
Row sums give A266475.
Cf. A000009, A000040, A001222, A112798, A246688, A265145.

Programs

  • Maple
    T:= n-> ((l-> seq(l[j]+j-1, j=1..nops(l)))(sort([seq(
       numtheory[pi](i[1])$i[2], i=ifactors(n)[2])]))):
seq(T(n), n=1..100);
  • Mathematica
    T[n_] := Function[l, Table[l[[j]]+j-1, {j, 1, Length[l]}]][Sort[ Flatten[ Table[ Array[ PrimePi[i[[1]]]&, i[[2]]], {i, FactorInteger[n]}]]]];
Table[T[n], {n, 1, 100}] // Flatten // Rest (* Jean-François Alcover, Mar 23 2017, translated from Maple *)

Formula

T(prime(n),1) = n.

A344090 Flattened tetrangle of strict integer partitions, sorted first by sum, then by length, then lexicographically.

Original entry on oeis.org

1, 2, 3, 2, 1, 4, 3, 1, 5, 3, 2, 4, 1, 6, 4, 2, 5, 1, 3, 2, 1, 7, 4, 3, 5, 2, 6, 1, 4, 2, 1, 8, 5, 3, 6, 2, 7, 1, 4, 3, 1, 5, 2, 1, 9, 5, 4, 6, 3, 7, 2, 8, 1, 4, 3, 2, 5, 3, 1, 6, 2, 1, 10, 6, 4, 7, 3, 8, 2, 9, 1, 5, 3, 2, 5, 4, 1, 6, 3, 1, 7, 2, 1, 4, 3, 2, 1
Offset: 0

Views

Author

Gus Wiseman, May 12 2021

Keywords

Comments

The zeroth row contains only the empty partition.
A tetrangle is a sequence of finite triangles.

Examples

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (3)(21)
  4: (4)(31)
  5: (5)(32)(41)
  6: (6)(42)(51)(321)
  7: (7)(43)(52)(61)(421)
  8: (8)(53)(62)(71)(431)(521)
  9: (9)(54)(63)(72)(81)(432)(531)(621)

Links

Crossrefs

Starting with reversed partitions gives A026793.
The version for compositions is A124734.
Showing partitions as Heinz numbers gives A246867.
The non-strict version is A334301 (reversed: A036036).
Ignoring length gives A344086 (reversed: A246688).
Same as A344089 with partitions reversed.
The version for revlex instead of lex is A344092.
A026791 reads off lexicographically ordered reversed partitions.
A080577 reads off reverse-lexicographically ordered partitions.
A112798 reads off reversed partitions by Heinz number.
A296150 reads off partitions by Heinz number.
Cf. A036037, A036043, A103921, A185974, A193073, A211992, A296774, A334302, A334433, A334435, A334438, A334439, A334440, A334441, A334442, A344091.

Programs

  • Mathematica
    Table[Sort[Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,10}]

A344087 Flattened tetrangle of strict integer partitions sorted first by sum, then colexicographically.

Original entry on oeis.org

1, 2, 2, 1, 3, 3, 1, 4, 4, 1, 3, 2, 5, 3, 2, 1, 5, 1, 4, 2, 6, 4, 2, 1, 6, 1, 5, 2, 4, 3, 7, 5, 2, 1, 4, 3, 1, 7, 1, 6, 2, 5, 3, 8, 6, 2, 1, 5, 3, 1, 8, 1, 4, 3, 2, 7, 2, 6, 3, 5, 4, 9, 4, 3, 2, 1, 7, 2, 1, 6, 3, 1, 5, 4, 1, 9, 1, 5, 3, 2, 8, 2, 7, 3, 6, 4, 10
Offset: 0

Views

Author

Gus Wiseman, May 11 2021

Keywords

Comments

The zeroth row contains only the empty partition.
A tetrangle is a sequence of finite triangles.

Examples

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (21)(3)
  4: (31)(4)
  5: (41)(32)(5)
  6: (321)(51)(42)(6)
  7: (421)(61)(52)(43)(7)
  8: (521)(431)(71)(62)(53)(8)
  9: (621)(531)(81)(432)(72)(63)(54)(9)

Links

Crossrefs

Positions of first appearances are A015724.
Triangle sums are A066189.
Taking revlex instead of colex gives A118457.
The not necessarily strict version is A211992.
Taking lex instead of colex gives A344086.
A026793 gives reversed strict partitions in A-S order (sum/length/lex).
A319247 sorts strict partitions by Heinz number.
A329631 sorts reversed strict partitions by Heinz number.
A344090 gives strict partitions in A-S order (sum/length/lex).
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080576, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A246688, A272020, A299755, A296774, A304038, A319247, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344088, A344089, A344091.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

Programs

  • Mathematica
    colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Table[Sort[Select[IntegerPartitions[n],UnsameQ@@#&],colex],{n,0,10}]

A344088 Flattened tetrangle of reversed strict integer partitions sorted first by sum, then colexicographically.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 3, 4, 2, 3, 1, 4, 5, 1, 2, 3, 2, 4, 1, 5, 6, 1, 2, 4, 3, 4, 2, 5, 1, 6, 7, 1, 3, 4, 1, 2, 5, 3, 5, 2, 6, 1, 7, 8, 2, 3, 4, 1, 3, 5, 4, 5, 1, 2, 6, 3, 6, 2, 7, 1, 8, 9, 1, 2, 3, 4, 2, 3, 5, 1, 4, 5, 1, 3, 6, 4, 6, 1, 2, 7, 3, 7, 2, 8, 1, 9, 10
Offset: 0

Views

Author

Gus Wiseman, May 12 2021

Keywords

Comments

The zeroth row contains only the empty partition.
A tetrangle is a sequence of finite triangles.

Examples

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (12)(3)
  4: (13)(4)
  5: (23)(14)(5)
  6: (123)(24)(15)(6)
  7: (124)(34)(25)(16)(7)
  8: (134)(125)(35)(26)(17)(8)
  9: (234)(135)(45)(126)(36)(27)(18)(9)

Links

Crossrefs

Positions of first appearances are A015724.
Triangle sums are A066189.
The non-strict version is A080576.
Taking lex instead of colex gives A246688 (non-reversed: A344086).
The non-reversed version is A344087.
Taking revlex instead of colex gives A344089 (non-reversed: A118457).
A026793 gives reversed strict partitions in A-S order (sum/length/lex).
A319247 sorts strict partitions by Heinz number.
A329631 sorts reversed strict partitions by Heinz number.
A344090 gives strict partitions in A-S order (sum/length/lex).
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A272020, A299755, A296774, A304038, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344091.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

Programs

  • Mathematica
    colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Table[Sort[Reverse/@Select[IntegerPartitions[n],UnsameQ@@#&],colex],{n,0,10}]

A339351 Irregular triangle read by rows in which row n lists the compositions (ordered partitions) of n into distinct parts in lexicographic order.

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Dec 01 2020

Keywords

Examples

			Triangle begins:
[1],
[2],
[1, 2], [2, 1], [3],
[1, 3], [3, 1], [4],
[1, 4], [2, 3], [3, 2], [4, 1], [5],
...

Links

Crossrefs

Cf. A026793, A066099, A097910 (row lengths), A118457, A228369, A246688, A304797 (row sums), A339178.

Programs

  • Mathematica
    Table[Sort[Join @@ Permutations /@ Select[IntegerPartitions[n], UnsameQ @@ # &], OrderedQ[PadRight[{#1, #2}]] &], {n, 8}] // Flatten

A344091 Flattened tetrangle of all finite multisets of positive integers sorted first by sum, then by length, then colexicographically.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, May 12 2021

Keywords

Comments

First differs from A334302 for partitions of 9.
The zeroth row contains only the empty partition.
A tetrangle is a sequence of finite triangles.

Examples

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)(11)
  3: (3)(12)(111)
  4: (4)(22)(13)(112)(1111)
  5: (5)(23)(14)(122)(113)(1112)(11111)
  6: (6)(33)(24)(15)(222)(123)(114)(1122)(1113)(11112)(111111)

Links

Crossrefs

The version for lex instead of colex is A036036.
Starting with reversed partitions gives A036037.
Ignoring length gives A211992 (reversed: A080576).
Same as A334301 with partitions reversed.
The version for revlex instead of colex is A334302.
The Heinz numbers of these partitions are A334433.
The strict case is A344089.
A026791 reads off lexicographically ordered reversed partitions.
A080577 reads off reverse-lexicographically ordered partitions.
A112798 reads off reversed partitions by Heinz number.
A193073 reads off lexicographically ordered partitions.
A296150 reads off partitions by Heinz number.
Cf. A000041, A026793, A124734, A185974, A228531, A246688, A296774, A334433, A334435, A334438, A334439, A334441, A334442, A344086, A344090.

Programs

  • Mathematica
    Table[Reverse/@Sort[IntegerPartitions[n]],{n,0,9}]

A344084 Concatenated list of all finite nonempty sets of positive integers sorted first by maximum, then by length, and finally lexicographically.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, May 11 2021

Keywords

Examples

			The sets are the columns below:
  1 2 1 3 1 2 1 4 1 2 3 1 1 2 1 5 1 2 3 4 1 1 1 2 2 3 1
      2   3 3 2   4 4 4 2 3 3 2   5 5 5 5 2 3 4 3 4 4 2
              3         4 4 4 3           5 5 5 5 5 5 3
                              4                       5
As a tetrangle, the first four triangles are:
  {1}
  {2},{1,2}
  {3},{1,3},{2,3},{1,2,3}
  {4},{1,4},{2,4},{3,4},{1,2,4},{1,3,4},{2,3,4},{1,2,3,4}

Links

Crossrefs

Triangle lengths are A000079.
Triangle sums are A001793.
Positions of first appearances are A005183.
Set maxima are A070939.
Set lengths are A124736.
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A026793, A036036, A036037, A048793, A066099, A080577, A112798, A118457, A124734, A162247, A193073, A211992, A228100, A228531, A246688, A272020, A296774, A299755, A304038, A319247, A329631, A334301, A334302, A334439, A334442, A335122, A344085, A344086, A344087, A344088, A344089.
Partition/composition applications: A036043, A049085, A115623, A129129, A185974, A238966, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

Programs

  • Mathematica
    SortBy[Rest[Subsets[Range[5]]],Last]

A339178 Irregular triangle read by rows in which row n lists the compositions (ordered partitions) of n into distinct parts in reverse lexicographic order.

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Nov 26 2020

Keywords

Examples

			Triangle begins:
[1],
[2],
[3], [2, 1], [1, 2],
[4], [3, 1], [1, 3],
[5], [4, 1], [3, 2], [2, 3], [1, 4],
...

Links

Crossrefs

Cf. A026793, A066099, A097910 (row lengths), A118457, A228369, A246688, A304797 (row sums).

Programs

  • Mathematica
    Table[Sort[Join @@ Permutations /@ Select[IntegerPartitions[n], UnsameQ @@ # &], OrderedQ[PadRight[{#2, #1}]] &], {n, 8}] // Flatten
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