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 1-8 of 8 results.

A383512 Heinz numbers of conjugate Wilf partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85
Offset: 1

Views

Author

Gus Wiseman, May 13 2025

Keywords

Comments

First differs from A364347 in having 130 and lacking 110.
First differs from A381432 in lacking 65 and 133.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

Examples

			The terms together with their prime indices begin:
     1: {}           17: {7}            35: {3,4}
     2: {1}          19: {8}            37: {12}
     3: {2}          20: {1,1,3}        38: {1,8}
     4: {1,1}        22: {1,5}          39: {2,6}
     5: {3}          23: {9}            40: {1,1,1,3}
     7: {4}          25: {3,3}          41: {13}
     8: {1,1,1}      26: {1,6}          43: {14}
     9: {2,2}        27: {2,2,2}        44: {1,1,5}
    10: {1,3}        28: {1,1,4}        45: {2,2,3}
    11: {5}          29: {10}           46: {1,9}
    13: {6}          31: {11}           47: {15}
    14: {1,4}        32: {1,1,1,1,1}    49: {4,4}
    15: {2,3}        33: {2,5}          50: {1,3,3}
    16: {1,1,1,1}    34: {1,7}          51: {2,7}

Links

Crossrefs

Partitions of this type are counted by A098859.
The conjugate version is A130091, complement A130092.
Including differences of 0 gives A325367, counted by A325324.
The strict case is A325388, counted by A320348.
The complement is A383513, counted by A336866.
Also requiring distinct multiplicities gives A383532, counted by A383507.
These are the positions of strict rows in A383534, or squarefree numbers in A383535.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A325349 counts partitions with distinct augmented differences, ranks A325366.
A383530 counts partitions that are not Wilf or conjugate Wilf, ranks A383531.
A383709 counts Wilf partitions with distinct augmented differences, ranks A383712.
Cf. A000720, A005117, A048768, A109298, A325325, A325351, A325355, A325368, A381431, A383506.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], UnsameQ@@DeleteCases[Differences[Prepend[prix[#],0]],0]&]

A383513 Heinz numbers of non conjugate Wilf partitions.

Original entry on oeis.org

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 70, 72, 78, 84, 90, 96, 102, 105, 108, 110, 114, 120, 126, 132, 133, 138, 140, 144, 147, 150, 154, 156, 162, 165, 168, 174, 180, 186, 189, 192, 198, 204, 210, 216, 220, 222, 228, 231, 234, 238, 240, 246
Offset: 1

Views

Author

Gus Wiseman, May 13 2025

Keywords

Comments

First differs from A381433 in having 65.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

Examples

			The terms together with their prime indices begin:
    6: {1,2}
   12: {1,1,2}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   30: {1,2,3}
   36: {1,1,2,2}
   42: {1,2,4}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   60: {1,1,2,3}
   63: {2,2,4}
   65: {3,6}
   66: {1,2,5}
   70: {1,3,4}
   72: {1,1,1,2,2}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}

Links

Crossrefs

Partitions of this type are counted by A336866.
The conjugate version is A130092, complement A130091.
Including differences of 0 gives complement of A325367, counted by A325324.
The strict case is the complement of A325388, counted by A320348.
The complement is A383512, counted by A098859.
Also forbidding distinct multiplicities gives A383531, counted by A383530.
These are positions of non-strict rows in A383534, or nonsquarefree numbers in A383535.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A383507 counts partitions that are Wilf and conjugate Wilf, ranks A383532.
A383709 counts Wilf partitions with distinct augmented differences, ranks A383712.
Cf. A000720, A005117, A048768, A238745, A325325, A325351, A325355, A325366, A325368, A381431, A383506.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!UnsameQ@@DeleteCases[Differences[Prepend[prix[#],0]],0]&]

A383709 Number of integer partitions of n with distinct multiplicities (Wilf) and distinct 0-appended differences.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 4, 4, 4, 5, 6, 5, 7, 8, 6, 8, 9, 9, 10, 9, 10, 12, 12, 11, 12, 14, 13, 14, 15, 14, 16, 16, 16, 18, 17, 17, 19, 20, 19, 19, 21, 21, 22, 22, 21, 24, 24, 23, 25, 25, 25, 26, 27, 27, 27, 28, 28, 30, 30, 28, 31, 32, 31, 32, 32, 33, 34, 34, 34
Offset: 0

Views

Author

Gus Wiseman, May 15 2025

Keywords

Comments

Integer partitions with distinct multiplicities are called Wilf partitions.

Examples

			The a(1) = 1 through a(8) = 4 partitions:
  (1)  (2)    (3)  (4)    (5)      (6)      (7)      (8)
       (1,1)       (2,2)  (3,1,1)  (3,3)    (3,2,2)  (4,4)
                                   (4,1,1)  (3,3,1)  (3,3,2)
                                            (5,1,1)  (6,1,1)

Crossrefs

For just distinct multiplicities we have A098859, ranks A130091, conjugate A383512.
For just distinct 0-appended differences we have A325324, ranks A325367.
For positive differences we have A383507, ranks A383532.
These partitions are ranked by A383712.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A239455 counts Look-and-Say partitions, complement A351293.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A383530 counts partitions that are not Wilf or conjugate-Wilf, ranks A383531.
A383534 gives 0-prepended differences by rank, see A325351.
Cf. A047966, A320348, A325325, A325349, A325368, A325388, A381431, A383506.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#]&&UnsameQ@@Differences[Append[#,0]]&]],{n,0,30}]

Formula

Ranked by A130091 /\ A325367

A383530 Number of non Wilf and non conjugate Wilf integer partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 3, 2, 5, 12, 14, 19, 35, 38, 55, 83, 107, 137, 209, 252, 359, 462, 612, 757, 1032, 1266, 1649, 2050, 2617, 3210, 4111, 4980, 6262, 7659, 9479, 11484, 14224, 17132, 20962, 25259, 30693, 36744, 44517, 53043, 63850, 75955, 90943, 107721, 128485
Offset: 0

Views

Author

Gus Wiseman, May 14 2025

Keywords

Comments

An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

Examples

			The a(0) = 0 through a(9) = 12 partitions:
  .  .  .  (21)  .  .  (42)    (421)   (431)    (63)
                       (321)   (3211)  (521)    (432)
                       (2211)          (3221)   (531)
                                       (4211)   (621)
                                       (32111)  (3321)
                                                (4221)
                                                (4311)
                                                (5211)
                                                (32211)
                                                (42111)
                                                (222111)
                                                (321111)

Links

Crossrefs

Negating both sides gives A383507, ranks A383532.
These partitions are ranked by A383531.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A239455 counts Look-and-Say partitions, complement A351293.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A381431 is the section-sum transform, union A381432, complement A381433.
A383534 gives 0-prepended differences by rank, see A325351.
A383709 counts Wilf partitions with distinct 0-appended differences, ranks A383712.
Cf. A033461, A047966, A111133, A320348, A325324, A325325, A325349, A325367, A325368, A325388, A383506.

Programs

  • Mathematica
    conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]], {k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Length/@Split[#]&&!UnsameQ@@Length/@Split[conj[#]]&]], {n,0,30}]

Formula

These partitions have Heinz numbers A130092 /\ A383513.

A383507 Number of Wilf and conjugate Wilf integer partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 7, 9, 12, 14, 19, 20, 27, 30, 31, 40, 50, 56, 68, 76, 86, 112, 126, 139, 170, 197, 216, 251, 297, 317, 378, 411, 466, 521, 607, 621, 745, 791, 892, 975, 1123, 1163, 1366, 1439, 1635, 1757, 2021, 2080, 2464, 2599, 2882, 3116, 3572, 3713
Offset: 0

Views

Author

Gus Wiseman, May 14 2025

Keywords

Comments

An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

Examples

			The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (1111)  (11111)  (222)     (331)      (332)
                                     (411)     (511)      (611)
                                     (3111)    (4111)     (2222)
                                     (111111)  (31111)    (5111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (11111111)

Links

Crossrefs

A048768 gives Look-and-Say fixed points, counted by A217605.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A239455 counts Look-and-Say partitions, complement A351293.
A325349 counts partitions with distinct augmented differences, ranks A325366.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A381431 is the section-sum transform, union A381432, complement A381433.
A383534 gives 0-prepended differences by rank, see A325351.
A383709 counts Wilf partitions with distinct 0-appended differences.
Cf. A047966, A048767, A111133, A320348, A325324, A325325, A325367, A325368, A325388, A351294, A383506.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#] && UnsameQ@@DeleteCases[Differences[Append[#,0]],0]&]],{n,0,30}]

Formula

These partitions have Heinz numbers A130091 /\ A383512.

A383535 Heinz number of the positive first differences of the 0-prepended prime indices of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 4, 7, 2, 3, 6, 11, 4, 13, 10, 6, 2, 17, 4, 19, 6, 9, 14, 23, 4, 5, 22, 3, 10, 29, 8, 31, 2, 15, 26, 10, 4, 37, 34, 21, 6, 41, 12, 43, 14, 6, 38, 47, 4, 7, 6, 33, 22, 53, 4, 15, 10, 39, 46, 59, 8, 61, 58, 9, 2, 25, 20, 67, 26, 51, 12, 71, 4, 73
Offset: 1

Views

Author

Gus Wiseman, May 21 2025

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
Also Heinz number of the first differences of the distinct 0-prepended prime indices of n.

Examples

			The terms together with their prime indices begin:
     1: {}        2: {1}        31: {11}       38: {1,8}
     2: {1}      17: {7}         2: {1}        47: {15}
     3: {2}       4: {1,1}      15: {2,3}       4: {1,1}
     2: {1}      19: {8}        26: {1,6}       7: {4}
     5: {3}       6: {1,2}      10: {1,3}       6: {1,2}
     4: {1,1}     9: {2,2}       4: {1,1}      33: {2,5}
     7: {4}      14: {1,4}      37: {12}       22: {1,5}
     2: {1}      23: {9}        34: {1,7}      53: {16}
     3: {2}       4: {1,1}      21: {2,4}       4: {1,1}
     6: {1,2}     5: {3}         6: {1,2}      15: {2,3}
    11: {5}      22: {1,5}      41: {13}       10: {1,3}
     4: {1,1}     3: {2}        12: {1,1,2}    39: {2,6}
    13: {6}      10: {1,3}      43: {14}       46: {1,9}
    10: {1,3}    29: {10}       14: {1,4}      59: {17}
     6: {1,2}     8: {1,1,1}     6: {1,2}       8: {1,1,1}

Crossrefs

For multiplicities instead of differences we have A181819.
Positions of first appearances are A358137.
Positions of squarefree numbers are A383512, counted by A098859.
Positions of nonsquarefree numbers are A383513, counted by A336866.
These are Heinz numbers of rows of A383534.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A320348 counts strict partitions with distinct 0-appended differences, ranks A325388.
A325324 counts partitions with distinct 0-appended differences, ranks A325367.
Cf. A122111, A124010 (prime signature), A130091, A287352, A325351, A325366, A325368, A355536, A381431, A384008, A384009.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@DeleteCases[Differences[Prepend[prix[n],0]],0],{n,100}]

Formula

A001222(a(n)) = A001221(n).
A056239(a(n)) = A061395(n).
A055396(a(n)) = A055396(n).
A061395(a(n)) = A241919(n).

A384009 Irregular triangle read by rows where row n lists the positive first differences of the prime indices of n.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 2, 2, 4, 1, 5, 3, 1, 1, 3, 6, 1, 1, 7, 4, 2, 1, 2, 4, 1, 8, 1, 2, 5, 5, 1, 2, 3, 6, 9, 1, 1, 10, 2, 3, 1, 3, 6, 7, 2, 1, 1, 11, 1, 7, 1, 1, 4, 2, 12, 1, 2, 4, 13, 8, 4, 1, 1, 2, 8, 9, 14, 5, 1, 3, 3, 2, 1, 5, 5, 1, 1, 15, 1, 2, 2, 10, 3, 1, 6, 6
Offset: 1

Views

Author

Gus Wiseman, May 23 2025

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 60 are {1,1,2,3}, differences (0,1,1), positive (1,1).
Rows begin:
     1: ()     16: ()       31: ()       46: (8)
     2: ()     17: ()       32: ()       47: ()
     3: ()     18: (1)      33: (3)      48: (1)
     4: ()     19: ()       34: (6)      49: ()
     5: ()     20: (2)      35: (1)      50: (2)
     6: (1)    21: (2)      36: (1)      51: (5)
     7: ()     22: (4)      37: ()       52: (5)
     8: ()     23: ()       38: (7)      53: ()
     9: ()     24: (1)      39: (4)      54: (1)
    10: (2)    25: ()       40: (2)      55: (2)
    11: ()     26: (5)      41: ()       56: (3)
    12: (1)    27: ()       42: (1,2)    57: (6)
    13: ()     28: (3)      43: ()       58: (9)
    14: (3)    29: ()       44: (4)      59: ()
    15: (1)    30: (1,1)    45: (1)      60: (1,1)

Crossrefs

Row-lengths are A001221(n) - 1, sums A243055.
For multiplicities instead of differences we have A124010 (prime signature).
Positions of non-strict rows are a subset of A325992.
Including difference 0 gives A355536, 0-prepended A287352.
The 0-prepended version is A383534.
A000040 lists the primes, differences A001223.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Cf. A039956, A048767, A055396, A061395, A130091, A320348, A325325, A325349, A325368, A358137, A381431.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[DeleteCases[Differences[prix[n]],0],{n,100}]

A384008 Irregular triangle read by rows where row n lists the first differences of the 0-prepended prime indices of the n-th squarefree number.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, May 23 2025

Keywords

Comments

All rows are different.

Examples

			The 28-th squarefree number is 42, with 0-prepended prime indices (0,1,2,4), with differences (1,1,2), so row 28 is (1,1,2).
The squarefree numbers and corresponding rows begin:
    1: ()        23: (9)        47: (15)
    2: (1)       26: (1,5)      51: (2,5)
    3: (2)       29: (10)       53: (16)
    5: (3)       30: (1,1,1)    55: (3,2)
    6: (1,1)     31: (11)       57: (2,6)
    7: (4)       33: (2,3)      58: (1,9)
   10: (1,2)     34: (1,6)      59: (17)
   11: (5)       35: (3,1)      61: (18)
   13: (6)       37: (12)       62: (1,10)
   14: (1,3)     38: (1,7)      65: (3,3)
   15: (2,1)     39: (2,4)      66: (1,1,3)
   17: (7)       41: (13)       67: (19)
   19: (8)       42: (1,1,2)    69: (2,7)
   21: (2,2)     43: (14)       70: (1,2,1)
   22: (1,4)     46: (1,8)      71: (20)

Crossrefs

Row-lengths are A072047, sums A243290.
This is the restriction of A383534 (ranked by A383535) to rows of squarefree index.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A320348 counts strict partitions with distinct 0-appended differences, ranks A325388.
A325324 counts partitions with distinct 0-appended differences, ranks A325367.
Cf. A001221, A005117, A055396, A061395, A325325, A355536, A358137, A384009.

Programs

  • Mathematica
    sql=Select[Range[100],SquareFreeQ];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Differences[Prepend[prix[sql[[n]]],0]],{n,Length[sql]}]
Showing 1-8 of 8 results.

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

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