A000701 -id:A000701 - cp's OEIS frontend

A352489 Weak excedance set of A122111. Numbers k <= A122111(k), where A122111 represents partition conjugation using Heinz numbers.

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 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, Mar 20 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The sequence lists all Heinz numbers of partitions whose Heinz number is less than or equal to that of their conjugate.

			The terms together with their prime indices begin:
   1: ()
   2: (1)
   3: (2)
   5: (3)
   6: (2,1)
   7: (4)
   9: (2,2)
  10: (3,1)
  11: (5)
  13: (6)
  14: (4,1)
  15: (3,2)
  17: (7)
  19: (8)
  20: (3,1,1)
For example, the partition (3,2,2) has Heinz number 45 and its conjugate (3,3,1) has Heinz number 50, and 45 <= 50, so 45 is in the sequence, and 50 is not.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
MathOverflow, Why 'excedances' of permutations? [closed].
Richard Ehrenborg and Einar Steingrímsson, The Excedance Set of a Permutation, Advances in Applied Mathematics 24, (2000), 284-299.

These partitions are counted by A046682.
The strong version is A352487, counted by A000701.
The opposite version is A352488, strong A352490
These are the positions of nonpositive terms in A352491.
A000041 counts integer partitions, strict A000009.
A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).
A003963 = product of prime indices, conjugate A329382.
A008292 is the triangle of Eulerian numbers (version without zeros).
A056239 adds up prime indices, row sums of A112798 and A296150.
A122111 = partition conjugation using Heinz numbers, parts A321649/A321650.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A173018 counts permutations by excedances, weak A123125.
A330644 counts non-self-conjugate partitions, ranked by A352486.
A352522 counts compositions by weak subdiagonals, rank statistic A352515.
Cf. A000720, A096276, A114088, A120383, A301987, A321648, A324850, A325040, A325044.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],#<=Times@@Prime/@conj[primeMS[#]]&]

a(n) <= A122111(a(n)).

A321728 Number of integer partitions of n whose Young diagram cannot be partitioned into vertical sections of the same sizes as the parts of the original partition.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 7, 10, 14, 20, 28, 37, 50

Offset: 0

Gus Wiseman, Nov 18 2018

First differs from A000701 at a(11) = 28, A000701(11) = 27
A vertical section is a partial Young diagram with at most one square in each row.
Conjecture: a(n) is the number of non-half-loop-graphical partitions of n. An integer partition is half-loop-graphical if it comprises the multiset of vertex-degrees of some graph with half-loops, where a half-loop is an edge with one vertex, to be distinguished from a full loop, which has two equal vertices.

			The a(2) = 1 through a(9) = 14 partitions whose Young diagram cannot be partitioned into vertical sections of the same sizes as the parts of the original partition are the same as the non-half-loop-graphical partitions up to n = 9:
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)
            (31)  (32)  (33)   (43)   (44)    (54)
                  (41)  (42)   (52)   (53)    (63)
                        (51)   (61)   (62)    (72)
                        (411)  (331)  (71)    (81)
                               (421)  (422)   (432)
                               (511)  (431)   (441)
                                      (521)   (522)
                                      (611)   (531)
                                      (5111)  (621)
                                              (711)
                                              (4311)
                                              (5211)
                                              (6111)
For example, a complete list of all half/full-loop-graphs with degrees y = (4,3,1) is the following:
  {{1,1},{1,2},{1,3},{2,2}}
  {{1},{2},{1,1},{1,2},{2,3}}
  {{1},{2},{1,1},{1,3},{2,2}}
  {{1},{3},{1,1},{1,2},{2,2}}
None of these is a half-loop-graph, as they have full loops (x,x), so y is counted under a(8).

Eric Weisstein's World of Mathematics, Degree Sequence.
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

The complement is counted by A321729.
Cf. A000110, A000258, A000700, A000701, A008277, A046682, A319616, A321730, A321737, A321738.
The following pertain to the conjecture.
Half-loop-graphical partitions by length are A029889 or A339843 (covering).
The version for full loops is A339655.
A027187 counts partitions of even length, with Heinz numbers A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A322661 counts labeled covering half-loop-graphs, ranked by A340018/A340019.
A339659 counts graphical partitions of 2n into k parts.
Cf. A006129, A025065, A062740, A095268, A096373, A167171, A320461, A338915, A339842, A339844, A339845.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Join@@Table[Subsets[ptnpos[y],{k}],{k,Reverse[Union[y]]}],UnsameQ@@First/@#&];
Table[Length[Select[IntegerPartitions[n],Select[spsu[ptnverts[#],ptnpos[#]],Function[p,Sort[Length/@p]==Sort[#]]]=={}&]],{n,8}]

a(n) is the number of integer partitions y of n such that the coefficient of m(y) in e(y) is zero, where m is monomial and e is elementary symmetric functions.

a(n) = A000041(n) - A321729(n).

A321729 Number of integer partitions of n whose Young diagram can be partitioned into vertical sections of the same sizes as the parts of the original partition.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 12, 16, 22, 28, 40, 51

Offset: 0

Gus Wiseman, Nov 18 2018

First differs from A046682 at a(11) = 28, A046682(11) = 29.
A vertical section is a partial Young diagram with at most one square in each row. For example, a suitable partition (shown as a coloring by positive integers) of the Young diagram of (322) is:
  1 2 3
  1 2
  2 3
Conjecture: a(n) is the number of half-loop-graphical partitions of n. An integer partition is half-loop-graphical if it comprises the multiset of vertex-degrees of some graph with half-loops, where a half-loop is an edge with one vertex, to be distinguished from a full loop, which has two equal vertices.

			The a(1) = 1 through a(8) = 12 partitions whose Young diagram cannot be partitioned into vertical sections of the same sizes as the parts of the original partition are the same as the half-loop-graphical partitions up to n = 8:
  (1)  (11)  (21)   (22)    (221)    (222)     (322)      (332)
             (111)  (211)   (311)    (321)     (2221)     (2222)
                    (1111)  (2111)   (2211)    (3211)     (3221)
                            (11111)  (3111)    (4111)     (3311)
                                     (21111)   (22111)    (4211)
                                     (111111)  (31111)    (22211)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
For example, the half-loop-graphs
  {{1},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3}}
both have degrees y = (3,2,2), so y is counted under a(7).

The complement is counted by A321728.
Cf. A000110, A000258, A000700, A000701, A006052, A007016, A008277, A046682, A319056, A319616, A321730, A321737, A321738.
The following pertain to the conjecture.
Half-loop-graphical partitions by length are A029889 or A339843 (covering).
The version for full loops is A339656.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A322661 counts labeled covering half-loop-graphs, ranked by A340018/A340019.
A339659 is a triangle counting graphical partitions by length.
Cf. A006129, A025065, A062740, A095268, A096373, A167171, A320461, A338915, A339842, A339844, A339845.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Join@@Table[Subsets[ptnpos[y],{k}],{k,Reverse[Union[y]]}],UnsameQ@@First/@#&];
Table[Length[Select[IntegerPartitions[n],Length[Select[spsu[ptnverts[#],ptnpos[#]],Function[p,Sort[Length/@p]==Sort[#]]]]>0&]],{n,8}]

a(n) is the number of integer partitions y of n such that the coefficient of m(y) in e(y) is nonzero, where m is monomial symmetric functions and e is elementary symmetric functions.

a(n) = A000041(n) - A321728(n).

A067619 Total number of parts in all self-conjugate partitions of n. Also, sum of largest parts of all self-conjugate partitions of n.

Original entry on oeis.org

0, 1, 0, 2, 2, 3, 3, 4, 7, 8, 9, 10, 15, 16, 18, 23, 30, 32, 35, 42, 51, 59, 63, 73, 89, 100, 106, 125, 145, 160, 174, 198, 229, 255, 274, 310, 355, 388, 420, 472, 534, 582, 631, 701, 784, 859, 928, 1021, 1144, 1243, 1338, 1475, 1630, 1767, 1909, 2089, 2299

Offset: 0

Naohiro Nomoto, Feb 01 2002

T. D. Noe, Table of n, a(n) for n = 0..1000
Arnold Knopfmacher and Neville Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9-18.

Cf. A000700, A000701, A006128, A015723, A046682, A079499, A158691.

CoefficientList[Series[Sum[n*q^(2n-1)*Product[1+q^k, {k, 1, 2n-3, 2}], {n, 1, 30}], {q, 0, 60}], q]

G.f.: A(q) = Sum_{n >= 1} n*q^(2*n-1)*(1+q)*(1+q^3)*...*(1+q^(2*n-3)).
From Peter Bala, Aug 20 2017: (Start)
Let F(q) = Product_{i >= 1} (1 + q^(2*i-1)). Then A(q) = Sum_{n >= 0} ( F(q) - Product_{i = 1..n} (1 + q^(2*i-1)) ).
It follows that the above sum A(q) satisfies -A(q-1) = 1 + q + 3*q^2 + 12*q^3 + 61*q^4 + ..., the g.f. for A158691, row-Fishburn matrices of size n. (End)

Edited by Dean Hickerson, Feb 11 2002

A238744 Irregular table read by rows: T (n, k) gives the number of primes p such that p^k divides n; table omits all zero values.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2

Offset: 2

Matthew Vandermast, Apr 28 2014

If the prime signature of n (nonincreasing version) is viewed as a partition, row n gives the conjugate partition.

			24 = 2^3*3 is divisible by two prime numbers (2 and 3), one square of a prime (4 = 2^2), and one cube of a prime (8 = 2^3); therefore, row 24 of the table is {2,1,1}.
From _Gus Wiseman_, Mar 31 2022: (Start)
Rows begin:
     1: ()        16: (1,1,1,1)    31: (1)
     2: (1)       17: (1)          32: (1,1,1,1,1)
     3: (1)       18: (2,1)        33: (2)
     4: (1,1)     19: (1)          34: (2)
     5: (1)       20: (2,1)        35: (2)
     6: (2)       21: (2)          36: (2,2)
     7: (1)       22: (2)          37: (1)
     8: (1,1,1)   23: (1)          38: (2)
     9: (1,1)     24: (2,1,1)      39: (2)
    10: (2)       25: (1,1)        40: (2,1,1)
    11: (1)       26: (2)          41: (1)
    12: (2,1)     27: (1,1,1)      42: (3)
    13: (1)       28: (2,1)        43: (1)
    14: (2)       29: (1)          44: (2,1)
    15: (2)       30: (3)          45: (2,1)
(End)

Row lengths are A051903(n); row sums are A001222(n).
Cf. A217171.
These partitions are ranked by A238745.
For prime indices A296150 instead of exponents we get A321649, rev A321650.
A000700 counts self-conjugate partitions, ranked by A088902.
A003963 gives product of prime indices, conjugate A329382.
A008480 gives number of permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798.
A124010 gives prime signature, sorted A118914, length A001221.
A352486-A352490 are sets related to the fixed points of A122111.
Cf. A000701, A000720, A046682, A238747, A258116, A319005, A330644, A352491.

Table[Length/@Table[Select[Last/@FactorInteger[n],#>=k&],{k,Max@@Last/@FactorInteger[n]}],{n,2,100}] (* Gus Wiseman, Mar 31 2022 *)

Row n is identical to row A124859(n) of table A212171.

A321732 Number of nonnegative integer square matrices with sum of entries equal to n, no zero rows or columns, and the same row sums as column sums.

Original entry on oeis.org

1, 1, 3, 11, 53, 317, 2293, 19435, 188851, 2068417, 25203807, 338117445, 4951449055, 78589443061, 1343810727205, 24626270763109, 481489261372381, 10004230113283129, 220125503239710879, 5113204953106107087, 125037079246130168973

Offset: 0

Gus Wiseman, Nov 18 2018

nonn

			The a(3) = 11 matrices:
  [3]
.
  [2 0] [1 1] [1 0] [0 1]
  [0 1] [1 0] [0 2] [1 1]
.
  [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]

Ludovic Schwob, Table of n, a(n) for n = 0..30

Cf. A000700, A000701, A006052, A007016, A120732, A319056, A320451, A321718, A321719, A321722, A321733, A321734, A321735, A321736, A321739.

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#]==Union[Last/@#],Total/@prs2mat[#]==Total/@Transpose[prs2mat[#]]]&]],{n,5}]

a(7) onwards from Ludovic Schwob, Apr 03 2024

A118302 Number of partitions of n into distinct parts with largest part not congruent to n modulo 2.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 3, 4, 5, 6, 8, 9, 11, 14, 16, 19, 23, 27, 32, 38, 45, 52, 61, 71, 82, 96, 111, 128, 148, 170, 195, 224, 256, 292, 334, 380, 432, 491, 556, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2048, 2291, 2560, 2859, 3189, 3554, 3959, 4404

Offset: 1

Reinhard Zumkeller, Apr 22 2006

nonn

a(2*n) = A026837(2*n), a(2*n+1) = A026838(2*n+1);
a(n) = A000009(n) - A118301(n),
a(A090864(n)) = A118303(n)/2 = A000009(A090864(n))/2.

			a(11) = #{10+1,8+3,8+2+1,6+5,6+4+1,6+3+2} = 6;
a(12) = #{11+1,9+3,9+2+1,7+5,7+4+1,7+3+2,5+4+3,5+4+2+1} = 8.

Cf. A000701, A118301.

Conjectural g.f.: A(x) = Limit_{N -> oo} ( Sum_{n = 1..2*N} (-1)^n/Product_{k = 1..n} 1 - x^(2*k-1) ). - Peter Bala, Feb 11 2021

A321730 Number of ways to partition the Young diagram of an integer partition of n into vertical sections of the same sizes as the parts of the original partition.

Original entry on oeis.org

1, 1, 1, 3, 8, 23, 79, 303, 1294, 5934, 29385, 156232, 884893

Offset: 0

Gus Wiseman, Nov 18 2018

A vertical section is a partial Young diagram with at most one square in each row. For example, a suitable partition (shown as a coloring by positive integers) of the Young diagram of (322) is:
2 3
2
3

			The a(5) = 23 partitions of Young diagrams of integer partitions of 5 into vertical sections of the same sizes as the parts of the original partition, shown as colorings by positive integers:
  1 2 3   1 2 3   1 2 3
  1       2       3
  1       2       3
.
  1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2
  1 2   1 3   1 3   2 1   3 1   3 1   2 3   3 2   2 3   3 2
  3     2     3     3     2     3     1     1     3     3
.
  1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2
  1     3     3     2     3     3     3     3     3
  3     1     4     3     2     4     3     4     4
  4     4     1     4     4     2     4     3     4
.
  1
  2
  3
  4
  5

Cf. A000110, A000258, A000700, A000701, A006052, A007016, A008277, A321728, A321729, A321731, A321737, A321738.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Join@@Table[Subsets[ptnpos[y],{k}],{k,Reverse[Union[y]]}],UnsameQ@@First/@#&];
Table[Sum[Length[Select[spsu[ptnverts[y],ptnpos[y]],Function[p,Sort[Length/@p]==Sort[y]]]],{y,IntegerPartitions[n]}],{n,5}]

A321731 Number of ways to partition the Young diagram of the integer partition with Heinz number n into vertical sections of the same sizes as the parts of the original partition.

Original entry on oeis.org

1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 0, 5, 0, 0, 0, 1, 0, 10, 0, 3, 0, 0, 0, 9, 0, 0, 8, 0, 0, 12, 0, 1, 0, 0, 0, 34, 0, 0, 0, 10, 0, 0, 0, 0, 24, 0, 0, 14, 0, 0, 0, 0, 0, 68, 0, 4, 0, 0, 0, 78, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 86, 0, 0, 36, 0, 0, 0, 0, 22, 60, 0, 0

Offset: 1

Gus Wiseman, Nov 18 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A vertical section is a partial Young diagram with at most one square in each row. For example, a suitable partition (shown as a coloring by positive integers) of the Young diagram of (322) is:
  1 2 3
  1 2
  2 3

			The a(30) = 12 partitions of the Young diagram of (321) into vertical sections of sizes (321), shown as colorings by positive integers:
  1 2 3   1 2 3   1 2 3   1 2 3   1 2 3   1 2 3
  1 2     1 3     2 1     3 1     1 2     1 3
  1       1       1       1       2       3
.
  1 2 3   1 2 3   1 2 3   1 2 3   1 2 3   1 2 3
  2 1     3 1     2 3     3 2     2 3     3 2
  2       3       2       2       3       3

Cf. A000110, A000258, A000700, A000701, A056239, A122111, A321649, A321728, A321729, A321730, A321737, A321738.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Join@@Table[Subsets[ptnpos[y],{k}],{k,Reverse[Union[y]]}],UnsameQ@@First/@#&];
Table[With[{y=Reverse[primeMS[n]]},Length[Select[spsu[ptnverts[y],ptnpos[y]],Function[p,Sort[Length/@p]==Sort[y]]]]],{n,30}]

A352143 Numbers whose prime indices and conjugate prime indices are all odd.

Original entry on oeis.org

1, 2, 5, 8, 11, 17, 20, 23, 31, 32, 41, 44, 47, 59, 67, 68, 73, 80, 83, 92, 97, 103, 109, 124, 125, 127, 128, 137, 149, 157, 164, 167, 176, 179, 188, 191, 197, 211, 227, 233, 236, 241, 257, 268, 269, 272, 275, 277, 283, 292, 307, 313, 320, 331, 332, 347, 353

Offset: 1

Gus Wiseman, Mar 18 2022

nonn

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, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
These are the Heinz numbers of integer partitions whose parts and conjugate parts are all odd. They are counted by A053253.

			The terms together with their prime indices begin:
   1: {}
   2: {1}
   5: {3}
   8: {1,1,1}
  11: {5}
  17: {7}
  20: {1,1,3}
  23: {9}
  31: {11}
  32: {1,1,1,1,1}
  41: {13}
  44: {1,1,5}
  47: {15}
  59: {17}
  67: {19}
  68: {1,1,7}
  73: {21}
  80: {1,1,1,1,3}

The restriction to primes is A031368.
These partitions appear to be counted by A053253.
The even version is A066207^2.
For even instead of odd conjugate parts we get A066208^2.
The first condition alone (all odd indices) is A066208, counted by A000009.
The second condition alone is A346635, counted by A000009.
A055922 counts partitions with odd multiplicities, ranked by A268335.
A066207 = indices all even, counted by A035363 (complement A086543).
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162642 counts odd prime exponents, even A162641.
A238745 gives the Heinz number of the conjugate prime signature.
A257991 counts odd indices, even A257992.
A258116 ranks strict partitions with all odd parts, even A258117.
A351979 = odd indices and even multiplicities, counted by A035457.
A352140 = even indices and odd multiplicities, counted by A055922 aerated.
A352141 = even indices and even multiplicities, counted by A035444.
A352142 = odd indices and odd multiplicities, counted by A117958.
Cf. A000290, A000701, A000720, A028260, A045931, A046682, A055396, A061395, A195017, A241638, A325698, A325700.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],And@@OddQ/@primeMS[#]&&And@@OddQ/@conj[primeMS[#]]&]

Intersection of A066208 and A346635.

cp's OEIS Frontend

A352489 Weak excedance set of A122111. Numbers k <= A122111(k), where A122111 represents partition conjugation using Heinz numbers.

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A321728 Number of integer partitions of n whose Young diagram cannot be partitioned into vertical sections of the same sizes as the parts of the original partition.

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A321729 Number of integer partitions of n whose Young diagram can be partitioned into vertical sections of the same sizes as the parts of the original partition.

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A067619 Total number of parts in all self-conjugate partitions of n. Also, sum of largest parts of all self-conjugate partitions of n.

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A238744 Irregular table read by rows: T (n, k) gives the number of primes p such that p^k divides n; table omits all zero values.

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A321732 Number of nonnegative integer square matrices with sum of entries equal to n, no zero rows or columns, and the same row sums as column sums.

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A118302 Number of partitions of n into distinct parts with largest part not congruent to n modulo 2.

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A321730 Number of ways to partition the Young diagram of an integer partition of n into vertical sections of the same sizes as the parts of the original partition.

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