A325329 -id:A325329 - cp's OEIS frontend

A325326 Heinz numbers of integer partitions covering an initial interval of positive integers with distinct multiplicities.

1, 2, 4, 8, 12, 16, 18, 24, 32, 48, 54, 64, 72, 96, 108, 128, 144, 162, 192, 256, 288, 324, 360, 384, 432, 486, 512, 540, 576, 600, 648, 720, 768, 864, 972, 1024, 1152, 1200, 1350, 1440, 1458, 1500, 1536, 1620, 1728, 1944, 2048, 2160, 2250, 2304, 2400, 2592

Offset: 1

Gus Wiseman, May 01 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A320348.

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     4: {1,1}
     8: {1,1,1}
    12: {1,1,2}
    16: {1,1,1,1}
    18: {1,2,2}
    24: {1,1,1,2}
    32: {1,1,1,1,1}
    48: {1,1,1,1,2}
    54: {1,2,2,2}
    64: {1,1,1,1,1,1}
    72: {1,1,1,2,2}
    96: {1,1,1,1,1,2}
   108: {1,1,2,2,2}
   128: {1,1,1,1,1,1,1}
   144: {1,1,1,1,2,2}
   162: {1,2,2,2,2}
   192: {1,1,1,1,1,1,2}
   256: {1,1,1,1,1,1,1,1}
   288: {1,1,1,1,1,2,2}
   324: {1,1,2,2,2,2}
   360: {1,1,1,2,2,3}
   384: {1,1,1,1,1,1,1,2}

Cf. A000837, A047966, A055932, A056239, A098859, A112798, A130091, A317081, A317089, A320348, A325329, A325337, A325369, A325372.

normQ[n_Integer]:=n==1||PrimePi/@First/@FactorInteger[n]==Range[PrimeNu[n]];
Select[Range[100],normQ[#]&&UnsameQ@@Last/@FactorInteger[#]&]

Intersection of normal numbers (A055932) and numbers with distinct prime exponents (A130091).

A319160 Number of integer partitions of n whose multiplicities appear with relatively prime multiplicities.

Original entry on oeis.org

1, 2, 2, 4, 5, 7, 11, 16, 22, 31, 45, 58, 83, 108, 142, 188, 250, 315, 417, 528, 674, 861, 1094, 1363, 1724, 2152, 2670, 3311, 4105, 5021, 6193, 7561, 9216, 11219, 13614, 16419, 19886, 23920, 28733, 34438, 41272, 49184, 58746, 69823, 82948, 98380, 116567

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

From Gus Wiseman, Jul 11 2023: (Start)
A partition is aperiodic (A000837) if its multiplicities are relatively prime. This sequence counts partitions whose multiplicities are aperiodic.
For example:
- The multiplicities of (5,3) are (1,1), with multiplicities (2), with common divisor 2, so it is not counted under a(8).
- The multiplicities of (3,2,2,1) are (2,1,1), with multiplicities (2,1), which are relatively prime, so it is counted under a(8).
- The multiplicities of (3,3,1,1) are (2,2), with multiplicities (2), with common divisor 2, so it is not counted under a(8).
- The multiplicities of (4,4,4,3,3,3,2,1) are (3,3,1,1), with multiplicities (2,2), which have common divisor 2, so it is not counted under a(24).
(End)

			The a(8) = 16 partitions:
  (8),
  (44),
  (332), (422), (611),
  (2222), (3221), (4211), (5111),
  (22211), (32111), (41111),
  (221111), (311111),
  (2111111),
  (11111111).
Missing from this list are: (53), (62), (71), (431), (521), (3311).

These partitions have ranks A319161.
For distinct instead of relatively prime multiplicities we have A325329.
Cf. A000837, A001597, A007916, A047966, A071625, A098859, A100953, A181819, A182850, A182857, A305563, A319149, A319162, A319164.

Table[Length[Select[IntegerPartitions[n], GCD@@Length/@Split[Sort[Length/@Split[#]]]==1&]],{n,30}]

A325369 Numbers with no two prime exponents appearing the same number of times in the prime signature.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 84, 85, 86

Offset: 1

Gus Wiseman, May 02 2019

nonn

The prime signature (A118914) is the multiset of exponents appearing in a number's prime factorization.

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. The enumeration of these partitions by sum is given by A325329.

			Most small numbers are in the sequence. However the sequence of non-terms together with their prime indices begins:
  12: {1,1,2}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  28: {1,1,4}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}
  50: {1,3,3}
  52: {1,1,6}
  54: {1,2,2,2}
  56: {1,1,1,4}
  63: {2,2,4}
  68: {1,1,7}
  72: {1,1,1,2,2}
  75: {2,3,3}
  76: {1,1,8}
  80: {1,1,1,1,3}
  88: {1,1,1,5}
For example, the prime indices of 1260 are {1,1,2,2,3,4}, whose multiplicities give the prime signature {1,1,2,2}, and since 1 and 2 appear the same number of times, 1260 is not in the sequence.

Cf. A056239, A098859, A112798, A118914, A130091, A317090, A319161, A325326, A325329, A325331, A325337, A325370, A325371.

Select[Range[100],UnsameQ@@Length/@Split[Sort[Last/@FactorInteger[#]]]&]

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

Gus Wiseman, May 01 2019

nonn

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.

			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).

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

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

Gus Wiseman, May 02 2019

nonn

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.

			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}

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

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

A325333 Number of integer partitions of n whose multiplicities all appear the same number of times.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 19, 23, 32, 39, 50, 63, 82, 96, 125, 152, 186, 226, 271, 326, 392, 473, 552, 663, 771, 918, 1065, 1261, 1448, 1710, 1953, 2283, 2608, 3062, 3455, 4013, 4552, 5271, 5974, 6884, 7774, 8937, 10065, 11570, 12953, 14838, 16710, 18979

Offset: 0

Gus Wiseman, May 01 2019

nonn

			The a(0) = 1 through a(7) = 14 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)
           (11)  (21)   (22)    (32)     (33)      (43)
                 (111)  (31)    (41)     (42)      (52)
                        (211)   (221)    (51)      (61)
                        (1111)  (311)    (222)     (322)
                                (2111)   (321)     (331)
                                (11111)  (411)     (421)
                                         (2211)    (511)
                                         (3111)    (2221)
                                         (21111)   (4111)
                                         (111111)  (22111)
                                                   (31111)
                                                   (211111)
                                                   (1111111)
For example, the partition (4,3,3,3,2,2,2,1) has multiplicities (1,3,3,1), and since both multiplicities 1 and 3 appear twice, (4,3,3,3,2,2,2,1) is counted under a(20).

Cf. A000041, A007862, A047966, A049988, A098859, A323022, A325329, A325369.

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

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A325326 Heinz numbers of integer partitions covering an initial interval of positive integers with distinct multiplicities.

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A319160 Number of integer partitions of n whose multiplicities appear with relatively prime multiplicities.

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A325369 Numbers with no two prime exponents appearing the same number of times in the prime signature.

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

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

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A325333 Number of integer partitions of n whose multiplicities all appear the same number of times.

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