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 1-9 of 9 results.

A325405 Heinz numbers of integer partitions y such that the k-th differences of y are distinct for all k >= 0 and are disjoint from the i-th differences for i != k.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 122
Offset: 1

Views

Author

Gus Wiseman, May 02 2019

Keywords

Comments

First differs from A325388 in lacking 130.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
The enumeration of these partitions by sum is given by A325404.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   19: {8}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   31: {11}
   33: {2,5}
   34: {1,7}
   35: {3,4}

Links

Crossrefs

A subsequence of A005117.
Cf. A056239, A112798, A279945, A325325, A325366, A325367, A325368, A325397, A325398, A325399, A325400, A325404, A325406, A325467.

Programs

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

A325393 Number of integer partitions of n whose k-th differences are strictly decreasing for all k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 8, 7, 9, 11, 10, 12, 15, 13, 16, 19, 18, 20, 24, 22, 26, 29, 28, 31, 37, 33, 38, 43, 42, 44, 52, 48, 55, 59, 58, 62, 72, 65, 74, 80, 80, 82, 94, 88, 99, 103, 104, 108, 123, 114, 126, 133, 135, 137, 155, 145, 161, 166, 169, 174
Offset: 0

Views

Author

Gus Wiseman, May 02 2019

Keywords

Comments

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
The Heinz numbers of these partitions are given by A325399.

Examples

			The a(1) = 1 through a(9) = 5 partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)   (8)    (9)
            (21)  (31)  (32)  (42)  (43)  (53)   (54)
                        (41)  (51)  (52)  (62)   (63)
                                    (61)  (71)   (72)
                                          (431)  (81)

Links

Crossrefs

Cf. A049988, A320466, A325353, A325354, A325358, A325391, A325396, A325399, A325404, A325406, A325457, A325468.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],And@@Table[Greater@@Differences[#,k],{k,0,Length[#]}]&]],{n,0,30}]

A325398 Heinz numbers of reversed integer partitions whose k-th differences are strictly increasing for all k >= 0.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109
Offset: 1

Views

Author

Gus Wiseman, May 02 2019

Keywords

Comments

First differs from A301899 in lacking 105. First differs from A325399 in having 42.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
The enumeration of these partitions by sum is given by A325391.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   19: {8}
   21: {2,4}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   31: {11}
   33: {2,5}

Links

Crossrefs

A subsequence of A005117.
Cf. A056239, A112798, A325357, A325391, A325395, A325397, A325399, A325400, A325405, A325406, A325456, A325467.

Programs

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

A325400 Heinz numbers of reversed integer partitions whose k-th differences are weakly increasing for all k >= 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74
Offset: 1

Views

Author

Gus Wiseman, May 02 2019

Keywords

Comments

First differs from A109427 in lacking 54.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
The enumeration of these partitions by sum is given by A325354.

Examples

			Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
   18: {1,2,2}
   36: {1,1,2,2}
   50: {1,3,3}
   54: {1,2,2,2}
   60: {1,1,2,3}
   70: {1,3,4}
   72: {1,1,1,2,2}
   75: {2,3,3}
   90: {1,2,2,3}
   98: {1,4,4}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  140: {1,1,3,4}
  144: {1,1,1,1,2,2}
  147: {2,4,4}
  150: {1,2,3,3}
  154: {1,4,5}
  162: {1,2,2,2,2}

Links

Crossrefs

Cf. A007294, A056239, A112798, A240026, A325354, A325360, A325362, A325394, A325397, A325398, A325399, A325405, A325467.

Programs

  • Mathematica
    primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],And@@Table[Greater@@Differences[primeptn[#],k],{k,0,PrimeOmega[#]}]&]

A325457 Heinz numbers of integer partitions with strictly decreasing differences.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98
Offset: 1

Views

Author

Gus Wiseman, May 03 2019

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The enumeration of these partitions by sum is given by A320470.

Examples

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   6: {1,2}
   7: {4}
   9: {2,2}
  10: {1,3}
  11: {5}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}

Links

Crossrefs

Cf. A056239, A112798, A320470, A320510, A325328, A325352, A325360, A325361, A325368, A325399, A325456, A325461, A320470, A325396.

Programs

  • Mathematica
    primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Greater@@Differences[primeptn[#]]&]

A325353 Number of integer partitions of n whose k-th differences are weakly decreasing for all k >= 0.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 7, 9, 11, 12, 13, 17, 16, 19, 23, 23, 24, 30, 29, 35, 37, 37, 40, 49, 47, 51, 56, 59, 61, 73, 65, 75, 80, 84, 91, 99, 91, 103, 112, 120, 114, 132, 126, 143, 154, 147, 152, 175, 169, 190, 187, 194, 198, 226, 225, 231, 236, 246, 256, 293
Offset: 0

Views

Author

Gus Wiseman, May 02 2019

Keywords

Comments

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
The Heinz numbers of these partitions are given by A325397.

Examples

			The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (321)     (2221)     (332)
                                     (111111)  (1111111)  (431)
                                                          (2222)
                                                          (11111111)
The first partition that has weakly decreasing differences (A320466) but is not counted under a(9) is (3,3,2,1), whose first and second differences are (0,-1,-1) and (-1,0) respectively.

Links

Crossrefs

Cf. A320466, A320509, A325350, A325354, A325391, A325393, A325397, A325398, A325399, A325404, A325405, A325406, A325468.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],And@@Table[GreaterEqual@@Differences[#,k],{k,0,Length[#]}]&]],{n,0,30}]

A325397 Heinz numbers of integer partitions whose k-th differences are weakly decreasing for all k >= 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 86, 87, 89
Offset: 1

Views

Author

Gus Wiseman, May 02 2019

Keywords

Comments

First differs from A325361 in lacking 150.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
The enumeration of these partitions by sum is given by A325353.

Examples

			Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   72: {1,1,1,2,2}
   76: {1,1,8}
   78: {1,2,6}
   80: {1,1,1,1,3}
The first partition that has weakly decreasing differences (A320466, A325361) but is not represented in this sequence is (3,3,2,1), which has Heinz number 150 and whose first and second differences are (0,-1,-1) and (-1,0) respectively.

Links

Crossrefs

Cf. A056239, A112798, A320466, A320509, A325353, A325361, A325364, A325389, A325398, A325399, A325400, A325405, A325467.

Programs

  • Mathematica
    primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],And@@Table[GreaterEqual@@Differences[primeptn[#],k],{k,0,PrimeOmega[#]}]&]

A325461 Heinz numbers of integer partitions with strictly decreasing differences (with the last part taken to be 0).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 67, 71, 73, 75, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197
Offset: 1

Views

Author

Gus Wiseman, May 03 2019

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) (with the last part taken to be 0) are (-3,-2,-1).
The enumeration of these partitions by sum is given by A320510.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   15: {2,3}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   29: {10}
   31: {11}
   35: {3,4}
   37: {12}
   41: {13}
   43: {14}

Links

Crossrefs

Cf. A056239, A112798, A320510, A325327, A325362, A325364, A325367, A325388, A325390, A325396, A325399, A325407, A325457, A325460.

Programs

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

A325467 Heinz numbers of integer partitions y such that the k-th differences of y are distinct (independently) for all k >= 0.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107
Offset: 1

Views

Author

Gus Wiseman, May 03 2019

Keywords

Comments

First differs from A301899 in having 70 and lacking 105.
First differs from A325398 in having 70.
First differs from A319315 in having 966.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
The enumeration of these partitions by sum is given by A325468.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   19: {8}
   21: {2,4}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   31: {11}
   33: {2,5}
For example, the k-th differences for k = 0...3 of the partition (9,4,2,1) with Heinz number 966 are
   9  4  2  1
  -5 -2 -1
   3  1
  -2
and since the entries of each row are distinct, 966 belongs to the sequence.

Links

Crossrefs

A subsequence of A005117.
Cf. A056239, A112798, A325366, A325367, A325368, A325397, A325398, A325399, A325400, A325405, A325468.

Programs

  • Mathematica
    primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],And@@Table[UnsameQ@@Differences[primeptn[#],k],{k,0,PrimeOmega[#]}]&]
Showing 1-9 of 9 results.

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