A304634 -id:A304634 - cp's OEIS frontend

A325268 Triangle read by rows where T(n,k) is the number of integer partitions of n with omicron k.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 0, 1, 0, 1, 5, 0, 0, 1, 0, 1, 7, 2, 0, 0, 1, 0, 1, 12, 1, 0, 0, 0, 1, 0, 1, 17, 2, 1, 0, 0, 0, 1, 0, 1, 24, 4, 0, 0, 0, 0, 0, 1, 0, 1, 33, 5, 1, 1, 0, 0, 0, 0, 1, 0, 1, 44, 9, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 57, 14, 3, 0, 1

Offset: 0

Gus Wiseman, Apr 18 2019

The omega-sequence of an integer partition is the sequence of lengths of the multisets obtained by repeatedly taking the multiset of multiplicities until a singleton is reached. The omicron of the partition is 0 if the omega-sequence is empty, 1 if it is a singleton, and otherwise the second-to-last part. For example, the partition (32211) has chain of multisets of multiplicities {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}, so its omega-sequence is (5,3,2,2,1), and its omicron is 2.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  3  0  1
  0  1  5  0  0  1
  0  1  7  2  0  0  1
  0  1 12  1  0  0  0  1
  0  1 17  2  1  0  0  0  1
  0  1 24  4  0  0  0  0  0  1
  0  1 33  5  1  1  0  0  0  0  1
  0  1 44  9  1  0  0  0  0  0  0  1
  0  1 57 14  3  0  1  0  0  0  0  0  1
  0  1 76 20  3  0  0  0  0  0  0  0  0  1
Row n = 8 counts the following partitions.
  (8)  (44)       (431)  (2222)  (11111111)
       (53)       (521)
       (62)
       (71)
       (332)
       (422)
       (611)
       (3221)
       (3311)
       (4211)
       (5111)
       (22211)
       (32111)
       (41111)
       (221111)
       (311111)
       (2111111)

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

Row sums are A000041. Column k = 2 is A325267.
Cf. A181819, A181821, A304634, A304636, A323014, A323023, A325250, A325273, A325277.
Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

Table[Length[Select[IntegerPartitions[n],Switch[#,{},0,{},1,,NestWhile[Sort[Length/@Split[#]]&,#,Length[#]>1&]//First]==k&]],{n,0,10},{k,0,n}]

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

A304687 Start with the multiset of prime multiplicities of n. Given a multiset, take the multiset of its multiplicities. Repeat until a constant multiset {k,k,...,k} is reached, and set a(n) to the sum of this multiset (k times the length).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 4, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 6, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 4, 2, 1, 2, 2, 2, 2

Offset: 1

Gus Wiseman, May 16 2018

nonn

			The following are examples showing the reduction of a multiset starting with the multiset of prime multiplicities of n.
         a(60) = 2: {1,1,2} -> {1,2} -> {1,1}.
        a(360) = 3: {1,2,3} -> {1,1,1}.
       a(1260) = 4: {1,1,2,2} -> {2,2}.
a(21492921450) = 6: {1,1,2,2,3,3} -> {2,2,2}.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A001221, A001222, A071625, A112798, A181819, A182850, A182857, A304465, A304634, A304636.

a:= proc(n) map(i-> i[2], ifactors(n)[2]);
      while nops({%[]})>1 do [coeffs(add(x^i, i=%))] od;
      add(i, i=%)
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 17 2018

Table[If[n==1,0,NestWhile[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],!SameQ@@#&]//Total],{n,360}]

A304636 Numbers n with prime omicron 3, meaning A304465(n) = 3.

Original entry on oeis.org

8, 27, 30, 42, 66, 70, 78, 102, 105, 110, 114, 125, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 343, 345, 354, 357, 360, 366, 370, 374, 385, 399, 402, 406, 410, 418, 426

Offset: 1

Gus Wiseman, May 15 2018

nonn

If n > 1 is not a prime number, we have A056239(n) >= Omega(n) >= omega(n) >= A071625(n) >= ... >= omicron(n) > 1 where Omega = A001222, omega = A001221, and omicron = A304465.

			This is a list of normalized factorizations (see A112798) of selected entries:
     8: {1,1,1}
    30: {1,2,3}
   360: {1,1,1,2,2,3}
   720: {1,1,1,1,2,2,3}
   900: {1,1,2,2,3,3}
  1440: {1,1,1,1,1,2,2,3}
  2160: {1,1,1,1,2,2,2,3}
  2880: {1,1,1,1,1,1,2,2,3}
  4320: {1,1,1,1,1,2,2,2,3}
  5760: {1,1,1,1,1,1,1,2,2,3}
  8640: {1,1,1,1,1,1,2,2,2,3}
Starting with A112798(1801800) and repeatedly taking the multiset of multiplicities we have {1,1,1,2,2,3,3,4,5,6} -> {1,1,1,2,2,3} -> {1,2,3} -> {1,1,1} -> {3}, so 1801800 belongs to the sequence.

Cf. A001221, A001222, A005117, A007916, A014612, A033992, A071625, A112798, A181819, A182850, A182857, A304464, A304465, A304634, A304647.

Join@@Position[Table[Switch[n,1,0,?PrimeQ,1,,NestWhile[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Length[#]>1&]//First],{n,200}],3]

A304647 Smallest term of A304636 that requires exactly n iterations to reach a fixed point under the x -> A181819(x) map.

Original entry on oeis.org

5, 8, 30, 360, 1801800, 2746644314348614680000, 13268350773236509446586539974366689358164301703214270074935844483572035447570761114173070859428708074413696096366645684575600000000

Offset: 0

Gus Wiseman, May 15 2018

nonn

The first entry 5 is optional so has offset 0.

			The list of multisets with Heinz numbers in the sequence is the following. The number of k's in row n + 1 is equal to the k-th largest term of row n.
                     5: {3}
                     8: {1,1,1}
                    30: {1,2,3}
                   360: {1,1,1,2,2,3}
               1801800: {1,1,1,2,2,3,3,4,5,6}
2746644314348614680000: {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,7,8,9,10}

Cf. A001221, A001222, A001462, A012257, A014612, A033992, A071625, A112798, A181819, A182850, A182857, A275870, A304464, A304465, A304634, A304636.

Function[m,Times@@Prime/@m]/@NestList[Join@@Table[Table[i,{Reverse[#][[i]]}],{i,Length[#]}]&,{3},6]

A325267 Number of integer partitions of n with omicron 2.

Original entry on oeis.org

0, 0, 1, 1, 3, 5, 7, 12, 17, 24, 33, 44, 57, 76, 100, 129, 168, 214, 282, 355, 462, 586, 755, 937, 1202, 1493, 1900, 2349, 2944, 3621, 4520, 5514, 6813, 8298, 10150, 12240, 14918, 17931, 21654, 25917, 31081, 37029, 44256, 52474, 62405, 73724, 87378, 102887

Offset: 0

Gus Wiseman, Apr 18 2019

nonn

The Heinz numbers of these partitions are given by A304634.

The omega-sequence of an integer partition is the sequence of lengths of the multisets obtained by repeatedly taking the multiset of multiplicities until a singleton is reached. We define the omicron of an integer partition to be 0 if the partition is empty, 1 if it is a singleton, and otherwise the second-to-last part of its omega-sequence. For example, the partition (32211) has chain of multisets of multiplicities {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}, so its omega-sequence is (5,3,2,2,1), and its omicron is 2.

			The a(1) = 1 through a(8) = 17 partitions:
  (11)  (21)  (22)   (32)    (33)     (43)      (44)
              (31)   (41)    (42)     (52)      (53)
              (211)  (221)   (51)     (61)      (62)
                     (311)   (411)    (322)     (71)
                     (2111)  (2211)   (331)     (332)
                             (3111)   (511)     (422)
                             (21111)  (2221)    (611)
                                      (3211)    (3221)
                                      (4111)    (3311)
                                      (22111)   (4211)
                                      (31111)   (5111)
                                      (211111)  (22211)
                                                (32111)
                                                (41111)
                                                (221111)
                                                (311111)
                                                (2111111)

Cf. A056239, A112798, A181819, A304634, A304636, A323014, A323023, A325250, A325273, A325277.
Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

Table[Length[Select[IntegerPartitions[n],Switch[#,{},0,{},1,,NestWhile[Sort[Length/@Split[#]]&,#,Length[#]>1&]//First]==2&]],{n,0,30}]

A319153 Number of integer partitions of n that reduce to 2, meaning their Heinz number maps to 2 under A304464.

Original entry on oeis.org

0, 2, 1, 3, 5, 7, 12, 17, 24, 33, 44, 57, 76, 100, 129, 168, 214, 282, 355, 462, 586, 755, 937, 1202, 1493, 1900, 2349, 2944, 3621, 4520, 5514, 6813, 8298, 10150, 12240, 14918, 17931, 21654, 25917, 31081, 37029, 44256, 52474, 62405, 73724, 87378, 102887

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

Start with an integer partition y of n. Given a multiset, take the multiset of its multiplicities. Repeat until a multiset of size 1 is obtained. If this multiset is {2}, we say that y reduces to 2. For example, we have (3211) -> (211) -> (21) -> (11) -> (2), so (3211) reduces to 2.

			The a(7) = 12 partitions:
  (43), (52), (61),
  (322), (331), (511),
  (2221), (3211), (4111),
  (22111), (31111),
  (211111).

Cf. A000837, A098859, A181819, A182850, A304464, A304465, A304634, A305563, A317245, A319149.

Table[Length[Select[IntegerPartitions[n],NestWhile[Sort[Length/@Split[#]]&,#,Length[#]>1&]=={2}&]],{n,30}]

cp's OEIS Frontend

A325268 Triangle read by rows where T(n,k) is the number of integer partitions of n with omicron k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A304687 Start with the multiset of prime multiplicities of n. Given a multiset, take the multiset of its multiplicities. Repeat until a constant multiset {k,k,...,k} is reached, and set a(n) to the sum of this multiset (k times the length).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A304636 Numbers n with prime omicron 3, meaning A304465(n) = 3.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A304647 Smallest term of A304636 that requires exactly n iterations to reach a fixed point under the x -> A181819(x) map.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A325267 Number of integer partitions of n with omicron 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A319153 Number of integer partitions of n that reduce to 2, meaning their Heinz number maps to 2 under A304464.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica