A109297 -id:A109297 - cp's OEIS frontend

A367682 Number of integer partitions of n whose multiset of multiplicities is the same as their multiset multiplicity kernel.

1, 1, 0, 1, 3, 2, 3, 2, 5, 5, 10, 9, 14, 14, 21, 20, 30, 36, 44, 50, 66, 75, 93, 106, 132, 151, 185, 212, 256, 286, 348, 394, 479, 543, 642, 740, 888, 994, 1176, 1350, 1589, 1789, 2109, 2371, 2786, 3144, 3653, 4126, 4811, 5385, 6213

Offset: 0

Gus Wiseman, Nov 30 2023

nonn

We define the multiset multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}. As an operation on multisets MMK is represented by A367579, and as an operation on their ranks it is represented by A367580.

			The a(1) = 1 through a(10) = 10 partitions:
  (1)  .  (21)  (22)   (41)   (51)    (61)   (71)     (81)    (91)
                (31)   (221)  (321)   (421)  (431)    (333)   (541)
                (211)         (3111)         (521)    (531)   (631)
                                             (3221)   (621)   (721)
                                             (41111)  (4221)  (3322)
                                                              (3331)
                                                              (4321)
                                                              (5221)
                                                              (322111)
                                                              (511111)

The case of strict partitions is A025147, ranks A039956.
The case of distinct multiplicities is A114640, ranks A109297.
These partitions have ranks A367683.
A000041 counts integer partitions, strict A000009.
A072233 counts partitions by number of parts.
A091602 counts partitions by greatest multiplicity, least A243978.
A116608 counts partitions by number of distinct parts.
Cf. A000837, A071625, A072774, A082090, A116861, A367579, A367580, A367581.

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[Length[Select[IntegerPartitions[n], Sort[Length/@Split[#]]==mmk[#]&]], {n,0,15}]

A367589 Numbers with exactly two distinct prime factors, both appearing with different exponents.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 72, 75, 76, 80, 88, 92, 96, 98, 99, 104, 108, 112, 116, 117, 124, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 200, 207, 208, 212, 224, 232, 236, 242, 244

Offset: 1

Gus Wiseman, Dec 01 2023

nonn

First differs from A177425 in lacking 360.
First differs from A182854 in lacking 360.
These are the Heinz numbers of the partitions counted by A182473.

			The terms together with their prime indices begin:
  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}

The case of any multiplicities is A007774, counts A002133.
These partitions are counted by A182473.
The case of equal exponents is A367590, counts A367588.
A000041 counts integer partitions, strict A000009.
A091602 counts partitions by greatest multiplicity, least A243978.
A098859 counts partitions with distinct multiplicities, ranks A130091.
A116608 counts partitions by number of distinct parts.
Cf. A071625, A072233, A072774, A109297, A367580.

Select[Range[100], PrimeNu[#]==2&&UnsameQ@@Last/@FactorInteger[#]&]

A367590 Numbers with exactly two distinct prime factors, both appearing with the same exponent.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194

Offset: 1

Gus Wiseman, Dec 01 2023

nonn

First differs from A268390 in lacking 210.
First differs from A238748 in lacking 210.
These are the Heinz numbers of the partitions counted by A367588.

			The terms together with their prime indices begin:
     6: {1,2}         57: {2,8}        106: {1,16}
    10: {1,3}         58: {1,10}       111: {2,12}
    14: {1,4}         62: {1,11}       115: {3,9}
    15: {2,3}         65: {3,6}        118: {1,17}
    21: {2,4}         69: {2,9}        119: {4,7}
    22: {1,5}         74: {1,12}       122: {1,18}
    26: {1,6}         77: {4,5}        123: {2,13}
    33: {2,5}         82: {1,13}       129: {2,14}
    34: {1,7}         85: {3,7}        133: {4,8}
    35: {3,4}         86: {1,14}       134: {1,19}
    36: {1,1,2,2}     87: {2,10}       141: {2,15}
    38: {1,8}         91: {4,6}        142: {1,20}
    39: {2,6}         93: {2,11}       143: {5,6}
    46: {1,9}         94: {1,15}       145: {3,10}
    51: {2,7}         95: {3,8}        146: {1,21}
    55: {3,5}        100: {1,1,3,3}    155: {3,11}

Michael De Vlieger, Table of n, a(n) for n = 1..10000

The case of any multiplicities is A007774, counts A002133.
Partitions of this type are counted by A367588.
The case of distinct exponents is A367589, counts A182473.
A000041 counts integer partitions, strict A000009.
A091602 counts partitions by greatest multiplicity, least A243978.
A116608 counts partitions by number of distinct parts.
Cf. A006881, A039956, A071625, A072233, A072774, A109297, A303661, A367580.

Select[Range[100], SameQ@@Last/@If[#==1, {}, FactorInteger[#]]&&PrimeNu[#]==2&]
Select[Range[200],PrimeNu[#]==2&&Length[Union[FactorInteger[#][[;;,2]]]]==1&] (* Harvey P. Dale, Aug 04 2025 *)

Union of A006881 and A303661. - Michael De Vlieger, Dec 01 2023

A367684 Number of integer partitions of n whose multiset multiplicity kernel is a submultiset.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 8, 10, 14, 17, 25, 30, 39, 51, 66, 79, 102, 125, 154, 191, 233, 284, 347, 420, 499, 614, 726, 867, 1031, 1233, 1437, 1726, 2002, 2375, 2770, 3271, 3760, 4455, 5123, 5994, 6904, 8064, 9199, 10753, 12241, 14202, 16189, 18704, 21194, 24504

Offset: 0

Gus Wiseman, Nov 30 2023

nonn

We define the multiset multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}. As an operation on multisets MMK is represented by A367579, and as an operation on their ranks it is represented by A367580.

			The a(1) = 1 through a(7) = 10 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (221)    (33)      (322)
                    (211)   (311)    (222)     (331)
                    (1111)  (2111)   (411)     (511)
                            (11111)  (2211)    (2221)
                                     (3111)    (4111)
                                     (21111)   (22111)
                                     (111111)  (31111)
                                               (211111)
                                               (1111111)

The case of strict partitions is A000012.
Includes all partitions with distinct multiplicities A098859, ranks A130091.
These partitions have ranks A367685.
A000041 counts integer partitions, strict A000009.
A072233 counts partitions by number of parts.
A091602 counts partitions by greatest multiplicity, least A243978.
A116608 counts partitions by number of distinct parts.
A116861 counts partitions by sum of distinct parts.
Cf. A000837, A032741, A071625, A109297, A114640, A367579, A367580, A367581.

submultQ[cap_,fat_]:=And@@Function[i, Count[fat,i]>=Count[cap, i]]/@Union[List@@cap];
mmk[q_List]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[Length[Select[IntegerPartitions[n], submultQ[mmk[#],#]&]], {n,0,15}]

A109299 Primal codes of canonical finite permutations on positive integers.

Original entry on oeis.org

1, 2, 12, 18, 360, 540, 600, 1350, 1500, 2250, 75600, 105840, 113400, 126000, 158760, 246960, 283500, 294000, 315000, 411600, 472500, 555660, 735000, 864360, 992250, 1296540, 1389150, 1440600, 1653750, 2572500, 3241350, 3601500, 3858750

Offset: 1

Jon Awbrey, Jul 09 2005

nonn

A canonical finite permutation on positive integers is a bijective mapping of [n] = {1, ..., n} to itself, counting the empty mapping as a permutation of the empty set.
From Rémy Sigrist, Sep 18 2021: (Start)
As usual with lists, the terms of the sequence are given in ascending order.
Equivalently, these are the numbers m such that A001221(m) = A051903(m) = A061395(m) = A071625(m).
This sequence has connections with A175061; here the prime factorizations, there the run-lengths in binary expansions, encode finite permutations.
There are m! terms with m distinct prime factors, the least one being A006939(m) and the greatest one being A076954(m); these m! terms are not necessarily contiguous. (End)

			Writing (prime(i))^j as i:j, we have this table:
Primal Codes of Canonical Finite Permutations
        1 = { }
        2 = 1:1
       12 = 1:2 2:1
       18 = 1:1 2:2
      360 = 1:3 2:2 3:1
      540 = 1:2 2:3 3:1
      600 = 1:3 2:1 3:2
     1350 = 1:1 2:3 3:2
     1500 = 1:2 2:1 3:3
     2250 = 1:1 2:2 3:3
    75600 = 1:4 2:3 3:2 4:1
   105840 = 1:4 2:3 3:1 4:2
   113400 = 1:3 2:4 3:2 4:1
   126000 = 1:4 2:2 3:3 4:1
   158760 = 1:3 2:4 3:1 4:2
   246960 = 1:4 2:2 3:1 4:3
   283500 = 1:2 2:4 3:3 4:1
   294000 = 1:4 2:1 3:3 4:2
   315000 = 1:3 2:2 3:4 4:1
   411600 = 1:4 2:1 3:2 4:3
   472500 = 1:2 2:3 3:4 4:1
   555660 = 1:2 2:4 3:1 4:3
   735000 = 1:3 2:1 3:4 4:2
   864360 = 1:3 2:2 3:1 4:4
   992250 = 1:1 2:4 3:3 4:2
  1296540 = 1:2 2:3 3:1 4:4
  1389150 = 1:1 2:4 3:2 4:3
  1440600 = 1:3 2:1 3:2 4:4
  1653750 = 1:1 2:3 3:4 4:2
  2572500 = 1:2 2:1 3:4 4:3
  3241350 = 1:1 2:3 3:2 4:4
  3601500 = 1:2 2:1 3:3 4:4
  3858750 = 1:1 2:2 3:4 4:3
  5402250 = 1:1 2:2 3:3 4:4

Suggested by Franklin T. Adams-Watters

J. Awbrey, Riffs and Rotes
Rémy Sigrist, PARI program for A109299

Cf. A001221, A006939, A051903, A061395, A071625, A076954, A106177, A108352, A108371, A109297, A109298, A109301, A175061, A347758.

PARI
```
\\ See Links section.
```

is(n) = { my (f=factor(n), p=f[,1]~, e=f[,2]~); Set(e)==[1..#e] && (#p==0 || p[#p]==prime(#p)) } \\ Rémy Sigrist, Sep 18 2021

Offset changed to 1 and data corrected by Rémy Sigrist, Sep 18 2021

A325767 Heinz numbers of integer partitions covering an initial interval of positive integers and containing their own multiset of multiplicities (as a submultiset).

Original entry on oeis.org

1, 2, 12, 18, 36, 60, 120, 180, 360, 450, 540, 600, 840, 1260, 1350, 1500, 1680, 1800, 2250, 2520, 2700, 3000, 3780, 4200, 4500, 5040, 5400, 5880, 6750, 8400, 9000, 10500, 11340, 11760, 12600, 13500, 15120, 17640, 18480, 18900, 20580, 21000, 22680, 25200

Offset: 1

Gus Wiseman, May 19 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 A325766.

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
    12: {1,1,2}
    18: {1,2,2}
    36: {1,1,2,2}
    60: {1,1,2,3}
   120: {1,1,1,2,3}
   180: {1,1,2,2,3}
   360: {1,1,1,2,2,3}
   450: {1,2,2,3,3}
   540: {1,1,2,2,2,3}
   600: {1,1,1,2,3,3}
   840: {1,1,1,2,3,4}
  1260: {1,1,2,2,3,4}
  1350: {1,2,2,2,3,3}
  1500: {1,1,2,3,3,3}
  1680: {1,1,1,1,2,3,4}
  1800: {1,1,1,2,2,3,3}
  2250: {1,2,2,3,3,3}
  2520: {1,1,1,2,2,3,4}

Cf. A000009, A055932, A056239, A112798, A109297, A114640, A290689, A325702, A325708, A325766.

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Select[Range[1000],#==1||Range[PrimeNu[#]]==PrimePi/@First/@FactorInteger[#]&&Divisible[#,red[#]]&]

A320252 Numbers with prime factorization Product_{k=1..w} prime(i_k) ^ e_k (where w = A001221(n) and prime(i) denotes the i-th prime number) such that i_k <> e_k for k = 1..w and { i_1, ..., i_w } = { e_1, ..., e_w }.

Original entry on oeis.org

1, 12, 40, 112, 352, 540, 600, 675, 832, 2176, 2268, 2352, 3969, 4864, 10692, 11616, 11776, 27440, 29403, 29696, 32448, 35000, 37908, 63488, 75600, 105840, 110976, 113400, 123201, 148716, 151552, 158760, 212960, 214375, 237600, 275000, 277248, 335872, 411600

Offset: 1

Rémy Sigrist, Oct 08 2018

nonn

This sequence is a subsequence of A109297.
For any i > 0 and j > 0 such that a(i) and a(j) are coprime, a(i) * a(j) belongs to this sequence.
For any i > 0, A048767(a(i)) belongs to this sequence.
Let S be the set of permutations of the natural numbers with finitely many non-fixed points:
- we can build a bijection f from S to this sequence as follows: for any s in S, f(s) = Product_{s(i) <> i} prime(i) ^ s(i),
- for any s in S with inverse z, f(z) = A048767(f(s)).

			The first terms, alongside the corresponding permutations, are:
  n   a(n)    s
  --  ------  ----------
   1       1  ()
   2      12  (1 2)
   3      40  (1 3)
   4     112  (1 4)
   5     352  (1 5)
   6     540  (1 2 3)
   7     600  (1 3 2)
   8     675  (2 3)
   9     832  (1 6)
  10    2176  (1 7)
  11    2268  (1 2 4)
  12    2352  (1 4 2)
  13    3969  (2 4)
  14    4864  (1 8)
  15   10692  (1 2 5)
  16   11616  (1 5 2)
  17   11776  (1 9)
  18   27440  (1 4 3)
  19   29403  (2 5)
  20   29696  (1 10)
  21   32448  (1 6 2)
  22   35000  (1 3 4)
  23   37908  (1 2 6)
  24   63488  (1 11)
  25   75600  (1 4)(2 3)

Cf. A001221, A048767, A071625, A109297.

is(n) = my (f=factor(n), i=apply(primepi, f[,1]~), e=f[,2]~); #select(k -> i[k]==e[k], [1..#f~])==0 && Set(i) == Set(e)

A001221(a(n)) = A071625(a(n)).

cp's OEIS Frontend

A367682 Number of integer partitions of n whose multiset of multiplicities is the same as their multiset multiplicity kernel.

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A367589 Numbers with exactly two distinct prime factors, both appearing with different exponents.

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A367590 Numbers with exactly two distinct prime factors, both appearing with the same exponent.

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A367684 Number of integer partitions of n whose multiset multiplicity kernel is a submultiset.

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Mathematica

A109299 Primal codes of canonical finite permutations on positive integers.

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PARI

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A325767 Heinz numbers of integer partitions covering an initial interval of positive integers and containing their own multiset of multiplicities (as a submultiset).

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Mathematica

A320252 Numbers with prime factorization Product_{k=1..w} prime(i_k) ^ e_k (where w = A001221(n) and prime(i) denotes the i-th prime number) such that i_k <> e_k for k = 1..w and { i_1, ..., i_w } = { e_1, ..., e_w }.

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