A317089 -id:A317089 - cp's OEIS frontend

A317090 Positive integers whose prime multiplicities span an initial interval of positive integers.

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85

Offset: 1

Gus Wiseman, Jul 21 2018

nonn

The first term in this sequence but absent from A179983 is 180.

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 6, 78, 820, 8379, 84440, 846646, 8473868, 84763404, 847714834, 8477408261, ... . Apparently, the asymptotic density of this sequence exists and equals 0.8477... . - Amiram Eldar, Aug 04 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index to sequences related to prime signature.

Cf. A001222, A007916, A055932, A056239, A069799, A112798, A124010, A133808.
Cf. A317087, A317088, A317089, A317091, A317092.
Subsequences: A129912\{1}, A179983\{1}.
Subsequence of A337533.

normalQ[m_]:=Union[m]==Range[Max[m]];
Select[Range[2,100],normalQ[FactorInteger[#][[All,2]]]&]

is(k) = {my(e = Set(factor(k)[,2])); k > 1 && vecmax(e) == #e;} \\ Amiram Eldar, Aug 04 2024

A317246 Heinz numbers of supernormal integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 30, 32, 60, 64, 90, 128, 150, 180, 210, 256, 300, 360, 450, 512, 540, 600, 1024, 1350, 1500, 2048, 2250, 2310, 2520, 3780, 4096, 4200, 5880, 8192, 9450, 10500, 12600, 13230, 15750, 16384, 17640, 18900, 20580, 26460, 29400, 30030

Offset: 1

Gus Wiseman, Jul 24 2018

nonn

An integer partition is supernormal if either (1) it is of the form 1^n for some n >= 0, or (2a) it spans an initial interval of positive integers, and (2b) its multiplicities, sorted in weakly decreasing order, are themselves a supernormal integer partition.

			Sequence of supernormal integer partitions begins: (), (1), (11), (21), (111), (211), (1111), (221), (321), (11111), (3211), (111111), (3221), (1111111), (3321), (32211), (4321).

Index entries for sequences related to Heinz numbers

Cf. A055932, A056239, A181819, A182850, A296150, A304465, A304687, A304818, A305732, A305733, A317089, A317090, A317245.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
supnrm[q_]:=Or[q=={}||Union[q]=={1},And[Union[q]==Range[Max[q]],supnrm[Sort[Length/@Split[q],Greater]]]];
Select[Range[10000],supnrm[primeMS[#]]&]

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[#]]&]

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

Original entry on oeis.org

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).

A332291 Heinz numbers of widely totally strongly normal integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 18, 30, 32, 64, 128, 210, 256, 450, 512, 1024, 2048, 2250, 2310, 4096, 8192, 16384, 30030, 32768, 65536, 131072, 262144, 510510, 524288

Offset: 1

Gus Wiseman, Feb 14 2020

An integer partition is widely totally strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) which are themselves a widely totally strongly normal partition.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
This sequence is closed under A304660, so there are infinitely many terms that are not powers of 2 or primorial numbers.

			The sequence of all widely totally strongly normal integer partitions together with their Heinz numbers begins:
      1: ()
      2: (1)
      4: (1,1)
      6: (2,1)
      8: (1,1,1)
     16: (1,1,1,1)
     18: (2,2,1)
     30: (3,2,1)
     32: (1,1,1,1,1)
     64: (1,1,1,1,1,1)
    128: (1,1,1,1,1,1,1)
    210: (4,3,2,1)
    256: (1,1,1,1,1,1,1,1)
    450: (3,3,2,2,1)
    512: (1,1,1,1,1,1,1,1,1)
   1024: (1,1,1,1,1,1,1,1,1,1)
   2048: (1,1,1,1,1,1,1,1,1,1,1)
   2250: (3,3,3,2,2,1)
   2310: (5,4,3,2,1)
   4096: (1,1,1,1,1,1,1,1,1,1,1,1)

Closed under A304660.
The non-strong version is A332276.
The co-strong version is A332293.
The case of reversed partitions is (also) A332293.
Heinz numbers of normal partitions with decreasing run-lengths are A025487.
Cf. A055932, A056239, A181819, A242031, A317089, A317246, A317257, A317492, A329747, A332277, A332278, A332290, A332292, A332297, A332337.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
totnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],GreaterEqual@@Length/@Split[ptn],totnQ[Length/@Split[ptn]]]];
Select[Range[10000],totnQ[Reverse[primeMS[#]]]&]

A317492 Heinz numbers of fully normal integer partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78

Offset: 1

Gus Wiseman, Jul 30 2018

nonn

An integer partition is fully normal if either it is of the form (1,1,...,1) or its multiplicities span an initial interval of positive integers and, sorted in weakly decreasing order, are themselves fully normal.

Cf. A055932, A056239, A181819, A182850, A296150, A305732, A317089, A317090, A317245, A317246, A317491, A317493.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
fulnrmQ[ptn_]:=With[{qtn=Sort[Length/@Split[ptn],Greater]},Or[ptn=={}||Union[ptn]=={1},And[Union[qtn]==Range[Max[qtn]],fulnrmQ[qtn]]]];
Select[Range[100],fulnrmQ[Reverse[primeMS[#]]]&]

A332276 Heinz numbers of widely totally normal integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 30, 32, 60, 64, 90, 128, 150, 180, 210, 256, 300, 360, 450, 512, 540, 600, 630, 1024, 1050, 1350, 1500, 2048, 2100, 2250, 2310, 2520, 2940, 3150, 3780, 4096, 4200, 4410, 5880, 8192, 8820, 9450, 10500, 11550, 12600, 13230, 14700

Offset: 1

Gus Wiseman, Feb 12 2020

nonn

First differs from A317246 in having 630.
A sequence of positive integers is widely totally normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has widely totally normal run-lengths.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}
   90: {1,2,2,3}
  128: {1,1,1,1,1,1,1}
  150: {1,2,3,3}
  180: {1,1,2,2,3}
  210: {1,2,3,4}
  256: {1,1,1,1,1,1,1,1}
  300: {1,1,2,3,3}
  360: {1,1,1,2,2,3}
For example, starting with (4,3,2,2,1), the partition with Heinz number 630, and repeatedly taking run-lengths gives (4,3,2,2,1) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1). These are all normal and the last is all 1's, so 630 belongs to the sequence.

Contains all powers of two A000079 and the primorials A002110.
Heinz numbers of normal integer partitions are A055932.
The case of reversed integer partitions is A332276 (this sequence).
The enumeration of these partitions by sum is A332277.
The enumeration of the generalization to compositions is A332279.
The co-strong version is A332290.
The strong version is A332291.
Cf. A007097, A056239, A133808, A181819, A182850, A305732, A317081, A317089, A317090, A317246, A317492, A329747, A332295, A332296.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
gnaQ[y_]:=Or[y=={},Union[y]=={1},And[Union[y]==Range[Max[y]],gnaQ[Length/@Split[y]]]];
Select[Range[1000],gnaQ[primeMS[#]]&]

A332290 Heinz numbers of widely alternately co-strongly normal integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 30, 32, 60, 64, 128, 210, 256, 360, 512, 1024, 2048, 2310, 2520, 4096, 8192, 16384, 30030, 32768, 65536, 75600, 131072, 262144, 510510, 524288

Offset: 1

Gus Wiseman, Feb 14 2020

An integer partition is widely alternately co-strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly increasing run-lengths (co-strong) which, if reversed, are themselves a widely alternately co-strongly normal partition.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
This sequence is closed under A181821, so there are infinitely many terms that are not powers of 2 or primorial numbers.

			The sequence of all widely alternately co-strongly normal integer partitions together with their Heinz numbers begins:
      1: ()
      2: (1)
      4: (1,1)
      6: (2,1)
      8: (1,1,1)
     12: (2,1,1)
     16: (1,1,1,1)
     30: (3,2,1)
     32: (1,1,1,1,1)
     60: (3,2,1,1)
     64: (1,1,1,1,1,1)
    128: (1,1,1,1,1,1,1)
    210: (4,3,2,1)
    256: (1,1,1,1,1,1,1,1)
    360: (3,2,2,1,1,1)
    512: (1,1,1,1,1,1,1,1,1)
   1024: (1,1,1,1,1,1,1,1,1,1)
   2048: (1,1,1,1,1,1,1,1,1,1,1)
   2310: (5,4,3,2,1)
   2520: (4,3,2,2,1,1,1)
For example, starting with y = (4,3,2,2,1,1,1), which has Heinz number 2520, and repeatedly taking run-lengths and reversing gives (4,3,2,2,1,1,1) -> (3,2,1,1) -> (2,1,1) -> (2,1) -> (1,1). These are all normal with weakly increasing run-lengths and the last is all 1's, so 2520 belongs to the sequence.

Closed under A181821.
The non-co-strong version is A332276.
The enumeration of these partitions by sum is A332289.
The total (rather than alternating) version is A332293.
Cf. A055932, A056239, A100883, A133808, A181819, A317089, A317090, A317246, A317257, A317492, A329747, A332292, A332340.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
totnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],LessEqual@@Length/@Split[ptn],totnQ[Reverse[Length/@Split[ptn]]]]];
Select[Range[10000],totnQ[Reverse[primeMS[#]]]&]

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

Gus Wiseman, May 02 2019

nonn

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.

			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.

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

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

A332293 Heinz numbers of widely totally co-strongly normal integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 30, 32, 64, 128, 180, 210, 256, 360, 512, 1024, 2048, 2310, 4096, 8192, 16384, 30030, 32768, 65536, 75600, 131072, 262144, 510510, 524288

Offset: 1

Gus Wiseman, Feb 16 2020

An integer partition is widely totally co-strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly increasing run-lengths (co-strong) which are themselves a widely totally co-strongly normal partition.

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

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
     8: {1,1,1}
    12: {1,1,2}
    16: {1,1,1,1}
    30: {1,2,3}
    32: {1,1,1,1,1}
    64: {1,1,1,1,1,1}
   128: {1,1,1,1,1,1,1}
   180: {1,1,2,2,3}
   210: {1,2,3,4}
   256: {1,1,1,1,1,1,1,1}
   360: {1,1,1,2,2,3}
   512: {1,1,1,1,1,1,1,1,1}
  1024: {1,1,1,1,1,1,1,1,1,1}
  2048: {1,1,1,1,1,1,1,1,1,1,1}
  2310: {1,2,3,4,5}
  4096: {1,1,1,1,1,1,1,1,1,1,1,1}
  8192: {1,1,1,1,1,1,1,1,1,1,1,1,1}
For example, 180 is the Heinz number of (3,2,2,1,1), with run-lengths (3,2,2,1,1) -> (1,2,2) -> (1,2) -> (1,1). These are all normal with weakly increasing multiplicities and the last is all 1's, so 180 belongs to the sequence.

A subset of A055932.
Closed under A181819.
The non-co-strong version is A332276.
The enumeration of these partitions by sum is A332278.
The alternating version is A332290.
The strong version is A332291.
The case of reversed partitions is (also) A332291.
Cf. A000009, A056239, A133808, A182850, A304660, A317089, A317246, A317257, A317492, A329747, A332277, A332289.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
gnaQ[y_]:=Or[y=={},Union[y]=={1},And[normQ[y],LessEqual@@Length/@Split[y],gnaQ[Length/@Split[y]]]];
Select[Range[1000],gnaQ[Reverse[primeMS[#]]]&]

cp's OEIS Frontend

A317090 Positive integers whose prime multiplicities span an initial interval of positive integers.

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A317246 Heinz numbers of supernormal integer partitions.

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

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

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Formula

A332291 Heinz numbers of widely totally strongly normal integer partitions.

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A317492 Heinz numbers of fully normal integer partitions.

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A332276 Heinz numbers of widely totally normal integer partitions.

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A332290 Heinz numbers of widely alternately co-strongly normal integer partitions.

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

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