A325334 -id:A325334 - cp's OEIS frontend

A325336 Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k whose parts cover an initial interval of positive integers.

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 0, 3, 1, 0, 0, 0, 0, 1, 0, 3, 2, 0, 0, 0, 0, 0, 1, 1, 3, 3, 0, 0, 0, 0, 0, 0, 1, 1, 5, 3, 0, 0, 0, 0, 0, 0, 0, 1, 0, 8, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 6, 6, 0, 0, 0

Offset: 0

Gus Wiseman, May 01 2019

The adjusted frequency depth of an integer partition (A325280) is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).

			Triangle begins:
  1
  0  1
  0  0  1
  0  0  1  1
  0  0  1  0  1
  0  0  1  0  2  0
  0  0  1  2  1  0  0
  0  0  1  0  3  1  0  0
  0  0  1  0  3  2  0  0  0
  0  0  1  1  3  3  0  0  0  0
  0  0  1  1  5  3  0  0  0  0  0
  0  0  1  0  8  3  0  0  0  0  0  0
  0  0  1  2  6  6  0  0  0  0  0  0  0
  0  0  1  0 13  4  0  0  0  0  0  0  0  0
  0  0  1  0 12  8  1  0  0  0  0  0  0  0  0
  0  0  1  2 14  7  3  0  0  0  0  0  0  0  0  0
  0  0  1  0 17 11  3  0  0  0  0  0  0  0  0  0  0
  0  0  1  0 22  7  8  0  0  0  0  0  0  0  0  0  0  0
  0  0  1  2 17 16 10  0  0  0  0  0  0  0  0  0  0  0  0
  0  0  1  0 28 10 15  0  0  0  0  0  0  0  0  0  0  0  0  0
  0  0  1  1 29 13 20  0  0  0  0  0  0  0  0  0  0  0  0  0  0
Row 15 counts the following partitions:
  111111111111111  54321       433221          333321        4322211
                   2222211111  443211          3332211       4332111
                               3322221         33222111      43221111
                               22222221        322221111
                               32222211        332211111
                               33321111        432111111
                               222222111       321111111111
                               3222111111
                               3321111111
                               22221111111
                               32211111111
                               222111111111
                               2211111111111
                               21111111111111

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A000009.
Column k = 3 is A325334.
Column k = 4 is A325335.
Cf. A181819, A182850, A317246, A320348, A323014, A325280, A325372.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];
Table[Length[Select[IntegerPartitions[n],normQ[#]&&fdadj[#]==k&]],{n,0,30},{k,0,n}]

depth(p)={if(!#p, 0, my(r=1); while(#p > 1, my(L=List(), k=0); for(i=1, #p, if(i==#p||p[i]<>p[i+1], listput(L,i-k); k=i)); listsort(L); p=L; r++); r)}
isok(p)={if(#p, for(i=1, #p, if(p[i]-1 > if(i>1, p[i-1], 0), return(0)))); 1}
row(n)={my(v=vector(1+n)); forpart(p=n, if(isok(p), v[1+depth(Vec(p))]++)); v}
{ for(n=0, 10, print(row(n))) } \\ Andrew Howroyd, Jan 18 2023

A325374 Numbers with adjusted frequency depth 3 whose prime indices cover an initial interval of positive integers.

Original entry on oeis.org

6, 30, 36, 210, 216, 900, 1296, 2310, 7776, 27000, 30030, 44100, 46656, 279936, 510510, 810000, 1679616, 5336100, 9261000, 9699690, 10077696, 24300000, 60466176, 223092870, 362797056, 729000000, 901800900, 1944810000, 2176782336, 6469693230, 12326391000

Offset: 1

Gus Wiseman, May 02 2019

nonn

The adjusted frequency depth (A323014) of a positive integer n is 0 if n = 1, and otherwise it is 1 plus the number of times one must apply A181819 to reach a prime number, where A181819(k = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of k. For example, 180 has adjusted frequency depth 5 because we have: 180 -> 18 -> 6 -> 4 -> 3.
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 adjusted frequency depth 3 whose parts cover an initial interval of positive integers. The enumeration of these partitions by sum is given by A325334.
The terms are the primorial numbers (A002110) above 2 and all their powers. - Amiram Eldar, May 08 2019

			The sequence of terms together with their prime indices begins:
      6: {1,2}
     30: {1,2,3}
     36: {1,1,2,2}
    210: {1,2,3,4}
    216: {1,1,1,2,2,2}
    900: {1,1,2,2,3,3}
   1296: {1,1,1,1,2,2,2,2}
   2310: {1,2,3,4,5}
   7776: {1,1,1,1,1,2,2,2,2,2}
  27000: {1,1,1,2,2,2,3,3,3}
  30030: {1,2,3,4,5,6}
  44100: {1,1,2,2,3,3,4,4}
  46656: {1,1,1,1,1,1,2,2,2,2,2,2}

Amiram Eldar, Table of n, a(n) for n = 1..5292

Cf. A002110, A055932, A056239, A112798, A130091, A181819, A320348, A323014, A325280, A325326, A325336, A325387.

normQ[n_Integer]:=Or[n==1,PrimePi/@First/@FactorInteger[n]==Range[PrimeNu[n]]];
fdadj[n_Integer]:=If[n==1,0,Length[NestWhileList[Times@@Prime/@Last/@FactorInteger[#1]&,n,!PrimeQ[#1]&]]];
Select[Range[10000],normQ[#]&&fdadj[#]==3&]

A325335 Number of integer partitions of n with adjusted frequency depth 4 whose parts cover an initial interval of positive integers.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 1, 3, 3, 3, 5, 8, 6, 13, 12, 14, 17, 22, 17, 28, 29, 30, 38, 50, 46, 67, 64, 75, 81, 104, 99, 127, 128, 150, 155, 201, 189, 236, 244, 293, 302, 363, 372, 437, 457, 548, 547, 638, 671, 754, 809, 922, 947, 1074, 1144, 1290, 1342, 1515, 1574

Offset: 0

Gus Wiseman, May 01 2019

nonn

The adjusted frequency depth of an integer partition (A325280) is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).

The Heinz numbers of these partitions are given by A325387.

			The a(4) = 1 through a(10) = 5 partitions:
  (211)  (221)   (21111)  (2221)    (22211)    (22221)     (222211)
         (2111)           (22111)   (221111)   (2211111)   (322111)
                          (211111)  (2111111)  (21111111)  (2221111)
                                                           (22111111)
                                                           (211111111)

Column k = 4 of A325336.

Cf. A000009, A007862, A181819, A182850, A317081, A320348, A323014, A325280, A325326, A325334, A325387.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];
Table[Length[Select[IntegerPartitions[n],normQ[#]&&fdadj[#]==4&]],{n,0,30}]

A325646 Number of separable partitions of n in which the number of distinct (repeatable) parts is 2.

Original entry on oeis.org

0, 0, 1, 2, 4, 4, 7, 8, 9, 11, 13, 14, 16, 18, 18, 22, 22, 25, 25, 29, 28, 32, 31, 38, 34, 39, 38, 44, 40, 49, 43, 52, 48, 53, 50, 63, 52, 60, 58, 69, 58, 73, 61, 74, 70, 74, 67, 90, 71, 84, 78, 89, 76, 97, 82, 100, 88, 95, 85, 119

Offset: 1

Clark Kimberling, May 16 2019

A partition is separable if there is an ordering of its parts in which no consecutive parts are identical. See A325534 for a guide to related sequences.

			a(6) counts these 4 partitions:  [5,1], [4,2], [1,4,1], [2,1,2,1].

Cf. A000041, A325334, A325647.

(separable=Table[Map[# [[1]]&,Select[Map[{#,Quotient[(1+Length[#]),Max[Map[Length,Split[#]]]]}&,IntegerPartitions[nn]],#[[2]]>1&]],{nn,35}]);
Map[Length[Select[Map[{#,Length[Union[#]]}&,#],#[[2]]==2&]]&,separable]
(* Peter J. C. Moses, May 08 2019 *)

A325712 Number of separable partitions of n in which the number of distinct (repeatable) parts <= 3.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 14, 19, 25, 35, 44, 56, 68, 87, 102, 124, 142, 171, 197, 225, 254, 294, 326, 370, 408, 451, 505, 553, 604, 661, 726, 772, 854, 916, 989, 1054, 1151, 1199, 1320, 1376, 1492, 1555, 1694, 1736, 1903, 1952, 2113, 2170, 2360, 2387, 2610

Offset: 1

Clark Kimberling, May 16 2019

A partition is separable if there is an ordering of its parts in which no consecutive parts are identical. See A325534 for a guide to related sequences.

Cf. A000041, A325334.

(separable=Table[Map[#[[1]]&,Select[Map[{#,Quotient[(1+Length[#]),Max[Map[Length,Split[#]]]]}&,IntegerPartitions[nn]],#[[2]]>1&]],{nn,35}]);
Map[Length[Select[Map[{#,Length[Union[#]]}&,#],#[[2]]<=3&]]&,separable]
(* Peter J. C. Moses, May 08 2019 *)

A325647 Number of separable partitions of n in which the number of distinct (repeatable) parts is 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 5, 9, 13, 21, 29, 39, 49, 68, 79, 101, 116, 145, 167, 196, 221, 262, 287, 335, 368, 412, 460, 512, 554, 617, 673, 723, 800, 865, 925, 1001, 1090, 1140, 1250, 1317, 1418, 1493, 1619, 1665, 1828, 1884, 2022, 2098, 2275, 2308, 2520, 2564

Offset: 1

Clark Kimberling, May 16 2019

A partition is separable if there is an ordering of its parts in which no consecutive parts are identical. See A325646 for a guide to related sequences.

			a(8) counts these 5 partitions:  [5,2,1], [4,3,1], [1,4,1,2], [2,3,2,1], [1,3,1,2,1].

Cf. A000041, A325334, A325646.

(separable=Table[Map[#[[1]]&,Select[Map[{#,Quotient[(1+Length[#]),Max[Map[Length,Split[#]]]]}&,IntegerPartitions[nn]],#[[2]]>1&]],{nn,35}]);
Map[Length[Select[Map[{#,Length[Union[#]]}&,#],#[[2]]==3&]]&,separable]
(* Peter J. C. Moses, May 08 2019 *)

A325711 Number of separable partitions of n in which the number of distinct (repeatable) parts <= 2.

Original entry on oeis.org

1, 1, 2, 3, 5, 5, 8, 9, 10, 12, 14, 15, 17, 19, 19, 23, 23, 26, 26, 30, 29, 33, 32, 39, 35, 40, 39, 45, 41, 50, 44, 53, 49, 54, 51, 64, 53, 61, 59, 70, 59, 74, 62, 75, 71, 75, 68, 91, 72, 85, 79, 90, 77, 98, 83, 101, 89, 96, 86, 120

Offset: 1

Clark Kimberling, May 16 2019

A partition is separable if there is an ordering of its parts in which no consecutive parts are identical. See A325534 for a guide to related sequences.

			a(6) counts these 5 partitions:  [6], [5,1], [4,2], [1,4,1], [2,1,2,1].

Cf. A000041, A325334, A325712.

(separable=Table[Map[#[[1]]&,Select[Map[{#,Quotient[(1+Length[#]),Max[Map[Length,Split[#]]]]}&,IntegerPartitions[nn]],#[[2]]>1&]],{nn,35}]);
Map[Length[Select[Map[{#,Length[Union[#]]}&,#],#[[2]]<=2&]]&,separable]
(* Peter J. C. Moses, May 08 2019 *)

A325713 Number of separable partitions of n in which the number of distinct (repeatable) parts <= 4.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 14, 19, 26, 37, 49, 66, 87, 115, 150, 193, 244, 309, 387, 479, 585, 714, 860, 1032, 1226, 1454, 1697, 1991, 2304, 2672, 3060, 3518, 3981, 4541, 5121, 5782, 6462, 7265, 8057, 9000, 9938, 11031, 12131, 13384, 14634, 16085, 17534, 19161

Offset: 1

Clark Kimberling, May 16 2019

A partition is separable if there is an ordering of its parts in which no consecutive parts are identical. See A325534 for a guide to related sequences.

Cf. A000041, A325334.

(separable=Table[Map[#[[1]]&,Select[Map[{#,Quotient[(1+Length[#]),Max[Map[Length,Split[#]]]]}&,IntegerPartitions[nn]],#[[2]]>1&]],{nn,35}]);
Map[Length[Select[Map[{#,Length[Union[#]]}&,#],#[[2]]<=4&]]&,separable]
(* Peter J. C. Moses, May 08 2019 *)

A325714 Number of separable partitions of n in which the number of distinct (repeatable) parts <= 5.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 14, 19, 26, 37, 49, 66, 87, 116, 152, 198, 254, 329, 422, 535, 676, 853, 1067, 1329, 1645, 2025, 2486, 3027, 3673, 4432, 5329, 6361, 7580, 8978, 10591, 12439, 14563, 16962, 19717, 22801, 26295, 30212, 34612, 39531, 45006, 51100, 57839

Offset: 1

Clark Kimberling, May 16 2019

A partition is separable if there is an ordering of its parts in which no consecutive parts are identical. See A325534 for a guide to related sequences.

Cf. A000041, A325334.

(separable=Table[Map[#[[1]]&,Select[Map[{#,Quotient[(1+Length[#]),Max[Map[Length,Split[#]]]]}&,IntegerPartitions[nn]],#[[2]]>1&]],{nn,35}]);
Map[Length[Select[Map[{#,Length[Union[#]]}&,#],#[[2]]<=5&]]&,separable]
(* Peter J. C. Moses, May 08 2019 *)

A325715 Number of separable partitions of n in which the number of distinct (repeatable) parts <= 6.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 14, 19, 26, 37, 49, 66, 87, 116, 152, 198, 254, 329, 422, 536, 678, 858, 1077, 1349, 1681, 2089, 2586, 3191, 3922, 4810, 5877, 7155, 8684, 10514, 12686, 15256, 18294, 21869, 26068, 30974, 36700, 43355, 51085, 59984, 70280, 82081, 95647

Offset: 1

Clark Kimberling, May 17 2019

A partition is separable if there is an ordering of its parts in which no consecutive parts are identical. See A325534 for a guide to related sequences.

Cf. A000041, A325334.

(separable=Table[Map[#[[1]]&,Select[Map[{#,Quotient[(1+Length[#]),Max[Map[Length,Split[#]]]]}&,IntegerPartitions[nn]],#[[2]]>1&]],{nn,35}]);
Map[Length[Select[Map[{#,Length[Union[#]]}&,#],#[[2]]<=6&]]&,separable]
(* Peter J. C. Moses, May 08 2019 *)

cp's OEIS Frontend

A325336 Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k whose parts cover an initial interval of positive integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A325374 Numbers with adjusted frequency depth 3 whose prime indices cover an initial interval of positive integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A325335 Number of integer partitions of n with adjusted frequency depth 4 whose parts cover an initial interval of positive integers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A325646 Number of separable partitions of n in which the number of distinct (repeatable) parts is 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A325712 Number of separable partitions of n in which the number of distinct (repeatable) parts <= 3.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A325647 Number of separable partitions of n in which the number of distinct (repeatable) parts is 3.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A325711 Number of separable partitions of n in which the number of distinct (repeatable) parts <= 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A325713 Number of separable partitions of n in which the number of distinct (repeatable) parts <= 4.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A325714 Number of separable partitions of n in which the number of distinct (repeatable) parts <= 5.

Views

Author

Keywords

Comments

Crossrefs