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 1-5 of 5 results.

A325337 Numbers whose prime exponents are distinct and cover an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 37, 41, 43, 44, 45, 47, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 101, 103, 107, 109, 113, 116, 117, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153, 157, 163, 164
Offset: 1

Views

Author

Gus Wiseman, May 01 2019

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions with distinct multiplicities covering an initial interval of positive integers. The enumeration of these partitions by sum is given by A320348.

Examples

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  11: {5}
  12: {1,1,2}
  13: {6}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  23: {9}
  28: {1,1,4}
  29: {10}
  31: {11}
  37: {12}
  41: {13}
  43: {14}
  44: {1,1,5}

Crossrefs

Cf. A055932, A056239, A112798, A317081, A317089, A317090, A325326, A325330, A325370, A325371.

Programs

  • Mathematica
    normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[100],UnsameQ@@Last/@FactorInteger[#]&&normQ[Last/@FactorInteger[#]]&]

A325370 Numbers whose prime signature has multiplicities covering an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 101, 103, 104
Offset: 1

Views

Author

Gus Wiseman, May 02 2019

Keywords

Comments

First differs from A319161 in lacking 420.
The prime signature (A118914) is the multiset of exponents appearing in a number's prime factorization.
Numbers whose prime signature covers an initial interval are given by A317090.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose multiplicities have multiplicities covering an initial interval of positive integers. The enumeration of these partitions by sum is given by A325330.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   12: {1,1,2}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
   23: {9}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
For example, the prime indices of 1890 are {1,2,2,2,3,4}, whose multiplicities give the prime signature {1,1,1,3}, and since this does not cover an initial interval (2 is missing), 1890 is not in the sequence.

Crossrefs

Cf. A000009, A055932, A056239, A112798, A118914, A317081, A317089, A317090, A319161, A325326, A325330, A325337, A325369, A325371.

Programs

  • Mathematica
    normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[100],normQ[Length/@Split[Sort[Last/@FactorInteger[#]]]]&]

A325329 Number of integer partitions of n whose multiplicities appear with distinct multiplicities.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 7, 13, 18, 25, 30, 52, 57, 81, 109, 140, 167, 230, 267, 354, 428, 532, 630, 815, 942, 1166, 1385, 1695, 1966, 2440, 2810, 3422, 4008, 4828, 5630, 6847, 7905, 9527, 11135, 13340, 15498, 18636, 21591, 25769, 30086, 35630, 41379, 49150, 56880
Offset: 0

Views

Author

Gus Wiseman, May 01 2019

Keywords

Comments

The Heinz numbers of these partitions are given by A325369.
Partitions whose parts appear with distinct multiplicities are counted by A098859, with Heinz numbers A130091.

Examples

			The a(0) = 1 through a(8) = 13 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
           (11)  (21)   (22)    (32)     (33)      (43)       (44)
                 (111)  (31)    (41)     (42)      (52)       (53)
                        (1111)  (11111)  (51)      (61)       (62)
                                         (222)     (421)      (71)
                                         (321)     (3211)     (431)
                                         (2211)    (1111111)  (521)
                                         (111111)             (2222)
                                                              (3221)
                                                              (3311)
                                                              (4211)
                                                              (32111)
                                                              (11111111)
For example, in (4,2,1,1), the multiplicities are 1 and 2, and 2 appears 1 time while 1 appears 2 times, so (4,2,1,1) is counted under a(8).

Crossrefs

Cf. A098859, A130091, A317081, A320348, A325326, A325330, A325331, A325333, A325337, A325369.

Programs

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

A325331 Number of integer partitions of n whose multiplicities appear with distinct multiplicities that cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 3, 7, 10, 14, 18, 30, 34, 44, 65, 73, 88, 110, 127, 155, 183, 202, 231, 277, 301, 339, 382, 430, 461, 551, 579, 681, 762, 896, 1010, 1255, 1406, 1752, 2061, 2555, 3001, 3783, 4437, 5512, 6611, 8056, 9539, 11668, 13692, 16515, 19435, 23098
Offset: 0

Views

Author

Gus Wiseman, May 01 2019

Keywords

Comments

Partitions with distinct multiplicities that cover an initial interval of positive integers are counted by A320348, with Heinz numbers A325337. Partitions whose multiplicities appear with distinct multiplicities are counted by A325329, with Heinz numbers A325369. Partitions whose multiplicities appear with multiplicities that cover an initial interval of positive integers of counted by A325330, with Heinz numbers A325370.
The Heinz numbers of these partitions are given by A325371.

Examples

			The a(0) = 1 through a(8) = 7 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
           (11)  (111)  (22)    (11111)  (33)      (3211)     (44)
                        (1111)           (222)     (1111111)  (2222)
                                         (111111)             (3221)
                                                              (4211)
                                                              (32111)
                                                              (11111111)
For example, the partition p = (5,5,4,3,3,3,2,2) has multiplicities (2,3,1,2), which appear with multiplicities (1,2,1), which cover an initial interval but are not distinct, so p is not counted under a(27). The partition q = (5,5,5,4,4,4,3,3,2,2,1,1) has multiplicities (3,3,2,2,2), which appear with multiplicities (3,2), which are distinct but do not cover an initial interval, so q is not counted under a(39). The partition r = (3,3,2,1,1) has multiplicities (2,1,2), which appear with multiplicities (1,2), which are distinct and cover an initial interval, so r is counted under a(10).

Crossrefs

Cf. A098859, A130091, A317081, A317090, A320348, A325329, A325330, A325337, A325369, A325370, A325371.

Programs

  • Mathematica
    normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[IntegerPartitions[n],normQ[Length/@Split[Sort[Length/@Split[#]]]]&&UnsameQ@@Length/@Split[Sort[Length/@Split[#]]]&]],{n,0,30}]

A325371 Numbers whose prime signature has multiplicities of its parts all distinct and covering an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 60, 61, 64, 67, 71, 73, 79, 81, 83, 84, 89, 90, 97, 101, 103, 107, 109, 113, 120, 121, 125, 126, 127, 128, 131, 132, 137, 139, 140, 149, 150, 151, 156, 157, 163
Offset: 1

Views

Author

Gus Wiseman, May 02 2019

Keywords

Comments

The first term that is not 1 or a prime power is 60.
The prime signature (A118914) is the multiset of exponents appearing in a number's prime factorization.
Numbers whose prime signature has distinct parts that cover an initial interval are given by A325337.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose multiplicities appear with distinct multiplicities that cover an initial interval of positive integers. The enumeration of these partitions by sum is given by A325331.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   37: {12}

Crossrefs

Cf. A055932, A056239, A098859, A112798, A118914, A130091, A317090, A325329, A325330, A325331, A325337, A325369, A325370.

Programs

  • Mathematica
    normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[100],normQ[Length/@Split[Sort[Last/@FactorInteger[#]]]]&&UnsameQ@@Length/@Split[Sort[Last/@FactorInteger[#]]]&]
Showing 1-5 of 5 results.

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

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