A325370 -id:A325370 - cp's OEIS frontend

A329139 Numbers whose prime signature is an aperiodic word.

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

Gus Wiseman, Nov 09 2019

nonn

First differs from A319161 in having 1260 = 2*2 * 3^2 * 5^1 * 7^1. First differs from A325370 in having 420 = 2^2 * 3^1 * 5^1 * 7^1.
A number's prime signature (A124010) is the sequence of positive exponents in its prime factorization.
A sequence is aperiodic if its cyclic rotations are all different.

			The sequence of terms together with their prime signatures begins:
   1: ()
   2: (1)
   3: (1)
   4: (2)
   5: (1)
   7: (1)
   8: (3)
   9: (2)
  11: (1)
  12: (2,1)
  13: (1)
  16: (4)
  17: (1)
  18: (1,2)
  19: (1)
  20: (2,1)
  23: (1)
  24: (3,1)
  25: (2)
  27: (3)

Complement of A329140.
Aperiodic compositions are A000740.
Aperiodic binary words are A027375.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose prime signature is a Lyndon word are A329131.
Numbers whose prime signature is a necklace are A329138.
Cf. A025487, A097318, A112798, A124010, A178472, A181819, A304678, A329133, A329135, A329136, A329137, A329142.

aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
Select[Range[100],aperQ[Last/@FactorInteger[#]]&]

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

Gus Wiseman, May 01 2019

nonn

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.

			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}

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

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

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[#]]]&]

A325330 Number of integer partitions of n whose multiplicities have multiplicities that cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 11, 16, 22, 31, 44, 55, 77, 96, 127, 158, 208, 251, 329, 400, 501, 610, 766, 915, 1141, 1368, 1677, 2005, 2454, 2913, 3553, 4219, 5110, 6053, 7300, 8644, 10376, 12238, 14645, 17216, 20504, 24047, 28501, 33336, 39373, 45871, 53926, 62745

Offset: 0

Gus Wiseman, May 01 2019

nonn

Partitions whose parts cover an initial interval of positive integers are counted by A000009, with Heinz numbers A055932. Partitions whose multiplicities cover an initial interval of positive integers are counted by A317081, with Heinz numbers A317090. Partitions whose parts and multiplicities both cover an initial interval of positive integers are counted by A317088, with Heinz numbers A317089. Partitions whose multiplicities at every depth cover an initial interval of positive integers are counted by A317245, with Heinz numbers A317246.

The Heinz numbers of these partitions are given by A325370.

			The a(0) = 1 through a(8) = 16 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
           (11)  (111)  (22)    (221)    (33)      (322)      (44)
                        (211)   (311)    (222)     (331)      (332)
                        (1111)  (2111)   (411)     (511)      (422)
                                (11111)  (3111)    (2221)     (611)
                                         (21111)   (3211)     (2222)
                                         (111111)  (4111)     (3221)
                                                   (22111)    (4211)
                                                   (31111)    (5111)
                                                   (211111)   (22211)
                                                   (1111111)  (32111)
                                                              (41111)
                                                              (221111)
                                                              (311111)
                                                              (2111111)
                                                              (11111111)
For example, the partition (5,5,4,3,3,3,2,2) has multiplicities (2,1,3,2) with multiplicities (1,2,1) which cover the initial interval {1,2}, so (5,5,4,3,3,3,2,2) is counted under a(27).

Cf. A000837, A055932, A317081, A317088, A317089, A317090, A317245, A320348, A325331, A325333, A325337, A325370.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[IntegerPartitions[n],normQ[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

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[#]]]&]

A325373 Composite totally abnormal numbers. Heinz numbers of non-singleton totally abnormal integer partitions.

Original entry on oeis.org

9, 25, 27, 49, 81, 100, 121, 125, 169, 196, 225, 243, 289, 343, 361, 441, 484, 529, 625, 676, 729, 841, 961, 1000, 1089, 1156, 1225, 1331, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2187, 2197, 2209, 2401, 2601, 2744, 2809, 3025, 3125, 3249, 3364, 3375, 3481

Offset: 1

Gus Wiseman, May 02 2019

nonn

The first term that is not a perfect power (A001597) is 11880, with prime indices {1,1,1,2,2,2,3,5} and prime signature {1,1,3,3}.
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. A number n is totally abnormal iff (1) the prime indices of n do not cover an initial interval of positive integers, and either (2a) n is prime, or (2b) the prime exponents (or prime signature) of n forms a totally abnormal integer partition, or, equivalently to (2b), A181819(n) is totally abnormal.
The enumeration of totally abnormal integer partitions by sum is given by A325332.

			The sequence of terms together with their prime indices begins:
     9: {2,2}
    25: {3,3}
    27: {2,2,2}
    49: {4,4}
    81: {2,2,2,2}
   100: {1,1,3,3}
   121: {5,5}
   125: {3,3,3}
   169: {6,6}
   196: {1,1,4,4}
   225: {2,2,3,3}
   243: {2,2,2,2,2}
   289: {7,7}
   343: {4,4,4}
   361: {8,8}
   441: {2,2,4,4}
   484: {1,1,5,5}
   529: {9,9}
   625: {3,3,3,3}
   676: {1,1,6,6}

Cf. A001597, A055932, A056239, A112798, A181819, A317089, A317090, A317246, A319152, A319810, A325332, A325370, A325372.

normQ[n_Integer]:=Or[n==1,PrimePi/@First/@FactorInteger[n]==Range[PrimeNu[n]]];
totabnQ[n_]:=And[!normQ[n],PrimeQ[n]||totabnQ[Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]]]];
Select[Range[10000],!PrimeQ[#]&&totabnQ[#]&]

cp's OEIS Frontend

A329139 Numbers whose prime signature is an aperiodic word.

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

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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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A325330 Number of integer partitions of n whose multiplicities have multiplicities that cover an initial interval of positive integers.

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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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Mathematica

A325373 Composite totally abnormal numbers. Heinz numbers of non-singleton totally abnormal integer partitions.

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