A096373 -id:A096373 - cp's OEIS frontend

A339740 Non-products of distinct primes or squarefree semiprimes.

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336

Offset: 1

Gus Wiseman, Dec 20 2020

nonn

Differs from A293243 and A212164 in having 1080, with prime indices {1,1,1,2,2,2,3} and factorization into distinct squarefree numbers 2*3*6*30.

			The sequence of terms together with their prime indices begins:
      4: {1,1}             80: {1,1,1,1,3}
      8: {1,1,1}           81: {2,2,2,2}
      9: {2,2}             88: {1,1,1,5}
     16: {1,1,1,1}         96: {1,1,1,1,1,2}
     24: {1,1,1,2}        104: {1,1,1,6}
     25: {3,3}            108: {1,1,2,2,2}
     27: {2,2,2}          112: {1,1,1,1,4}
     32: {1,1,1,1,1}      121: {5,5}
     40: {1,1,1,3}        125: {3,3,3}
     48: {1,1,1,1,2}      128: {1,1,1,1,1,1,1}
     49: {4,4}            135: {2,2,2,3}
     54: {1,2,2,2}        136: {1,1,1,7}
     56: {1,1,1,4}        144: {1,1,1,1,2,2}
     64: {1,1,1,1,1,1}    152: {1,1,1,8}
     72: {1,1,1,2,2}      160: {1,1,1,1,1,3}
For example, a complete list of strict factorizations of 72 is: (2*3*12), (2*4*9), (2*36), (3*4*6), (3*24), (4*18), (6*12), (8*9), (72); but since none of these consists of only primes or squarefree semiprimes, 72 is in the sequence.

A013929 allows only primes.
A320894 does not allow primes (but omega is assumed even).
A339741 is the complement.
A339742 has zeros at these positions.
A339840 allows squares of primes.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A320663 counts non-isomorphic multiset partitions into singletons or pairs.
A339841 have exactly one factorization into primes or semiprimes.
The following count factorizations:
- A001055 into all positive integers > 1.
- A050326 into distinct squarefree numbers.
- A320655 into semiprimes.
- A320656 into squarefree semiprimes.
- A320732 into primes or semiprimes.
- A322353 into distinct semiprimes.
- A339661 into distinct squarefree semiprimes.
- A339839 into distinct primes or semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A058696 counts partitions of 2n (A300061).
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339617 counts non-graphical partitions of 2n (A339618).
- A321728 is conjectured to count non-half-loop-graphical partitions of n.
The following count partitions/factorizations of even length and give their Heinz numbers:
- A027187/A339846 counts all of even length (A028260).
- A096373/A339737 cannot be partitioned into strict pairs (A320891).
- A338915/A339662 cannot be partitioned into distinct pairs (A320892).
- A339559/A339564 cannot be partitioned into distinct strict pairs (A320894).
Cf. A001221, A005117, A050320, A320893, A320911, A320912, A320922, A320924, A339113, A339561.

sqps[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqps[n/d],Min@@#>d&]],{d,Select[Divisors[n],PrimeQ[#]||SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
Select[Range[100],sqps[#]=={}&]

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).

A339843 Number of distinct sorted degree sequences among all n-vertex half-loop-graphs without isolated vertices.

Original entry on oeis.org

1, 1, 3, 9, 29, 97, 336, 1188, 4275, 15579, 57358, 212908, 795657, 2990221, 11291665, 42814783, 162920417, 621885767, 2380348729

Offset: 0

Gus Wiseman, Dec 27 2020

In the covering case, these degree sequences, sorted in decreasing order, are the same thing as half-loop-graphical partitions (A321729). 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.
The following are equivalent characteristics for any positive integer n:
(1) the prime indices of n can be partitioned into distinct singletons or strict pairs, i.e., into a set of half-loops or edges;
(2) n can be factored into distinct primes or squarefree semiprimes;
(3) the prime signature of n is half-loop-graphical.

			The a(0) = 1 through a(3) = 9 sorted degree sequences:
  ()  (1)  (1,1)  (1,1,1)
           (2,1)  (2,1,1)
           (2,2)  (2,2,1)
                  (2,2,2)
                  (3,1,1)
                  (3,2,1)
                  (3,2,2)
                  (3,3,2)
                  (3,3,3)
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(3).

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

See link for additional cross references.
The version for simple graphs is A004251, covering: A095268.
The non-covering version (it allows isolated vertices) is A029889.
The same partitions counted by sum are conjectured to be A321729.
These graphs are counted by A006125 shifted left, covering: A322661.
The version for full loops is A339844, covering: A339845.
These graphs are ranked by A340018 and A340019.
A006125 counts labeled simple graphs, covering: A006129.
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.
A339659 counts graphical partitions of 2n into k parts.
Cf. A062740, A096373, A167171, A320461, A320893, A320921, A320923, A338915, A339560, A339841, A339842.

Table[Length[Union[Sort[Table[Count[Join@@#,i],{i,n}]]&/@Select[Subsets[Subsets[Range[n],{1,2}]],Union@@#==Range[n]&]]],{n,0,5}]

a(n) = A029889(n) - A029889(n-1) for n > 0. - Andrew Howroyd, Jan 10 2024

a(7)-a(18) added (using A029889) by Andrew Howroyd, Jan 10 2024

A097091 Number of partitions of n such that the least part occurs exactly three times.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 2, 2, 6, 5, 8, 11, 15, 18, 27, 30, 43, 54, 69, 83, 113, 134, 172, 211, 265, 320, 405, 483, 602, 726, 888, 1064, 1306, 1554, 1884, 2248, 2707, 3213, 3860, 4560, 5446, 6435, 7638, 8990, 10651, 12494, 14734, 17260, 20277, 23683, 27754, 32328

Offset: 1

Robert G. Wilson v, Jul 24 2004

nonn

Number of partitions p of n such that 2*min(p) + (number of parts of p) is a part of p. - Clark Kimberling, Feb 28 2014

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A002865, A096373, A097092, A097093.

a[n_] := Module[{p = IntegerPartitions[n], l = PartitionsP[n], c = 0, k = 1}, While[k < l + 1, q = PadLeft[p[[k]], 4]; If[q[[1]] != q[[4]] && q[[2]] == q[[4]], c++]; k++]; c]; Table[ a[n], {n, 52}]
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Length[p] + 2*Min[p]]], {n, 50}] (* Clark Kimberling, Feb 28 2014 *)
nmax = 60; Rest[CoefficientList[Series[Sum[x^(3*m)/Product[1-x^k,{k,m+1,nmax}], {m, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 04 2025 *)
Table[-PartitionsP[n] + 3 PartitionsP[3 + n] - PartitionsP[4 + n] - 2 PartitionsP[5 + n] + PartitionsP[6 + n], {n, 1, 60}] (* Vaclav Kotesovec, Jul 05 2025 *)

G.f.: Sum_{m>0} (x^(3*m) / Product_{i>m} (1-x^i)). More generally, g.f. for number of partitions of n such that the least part occurs exactly k times is Sum_{m>0} (x^(k*m) / Product_{i>m} (1-x^i)). Vladeta Jovovic
From Vaclav Kotesovec, Jul 05 2025: (Start)
a(n) = -p(n) + 3*p(n+3) - p(n+4) - 2*p(n+5) + p(n+6), where p(n) = A000041(n).
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)) * (1 - (3^(3/2)/(Pi*sqrt(2)) + 37*Pi/(24*sqrt(6)))/sqrt(n)). (End)

A097092 Number of partitions of n such that the least part occurs exactly four times.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 3, 2, 4, 5, 9, 9, 14, 16, 26, 29, 40, 48, 67, 79, 105, 126, 165, 196, 253, 303, 385, 459, 572, 687, 852, 1014, 1244, 1482, 1807, 2145, 2595, 3075, 3701, 4375, 5231, 6170, 7350, 8641, 10247, 12025, 14201, 16620, 19557, 22839, 26790, 31209

Offset: 1

Robert G. Wilson v, Jul 24 2004

nonn

Number of partitions p of n such that 3*min(p) + (number of parts of p) is a part of p. - Clark Kimberling, Feb 28 2014

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A002865, A096373, A097091, A097093.

a[n_] := Module[{p = IntegerPartitions[n], l = PartitionsP[n], c = 0, k = 1}, While[k < l + 1, q = PadLeft[ p[[k]], 5]; If[ q[[1]] != q[[5]] && q[[2]] == q[[5]], c++ ]; k++ ]; c]; Table[ a[n], {n, 53}]
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Length[p] + 3*Min[p]]], {n, 50}] (* Clark Kimberling, Feb 28 2014 *)
Table[Count[IntegerPartitions[n],?(Length[Split[#][[-1]]]==4&)],{n,60}] (* _Harvey P. Dale, Jan 18 2021 *)
nmax = 60; Rest[CoefficientList[Series[Sum[x^(4*m)/Product[1-x^k,{k,m+1,nmax}], {m, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 04 2025 *)
Table[-PartitionsP[n] + 4 PartitionsP[4 + n] - PartitionsP[5 + n] - 2 PartitionsP[6 + n] - 2 PartitionsP[7 + n] + PartitionsP[8 + n] + 2 PartitionsP[9 + n] - PartitionsP[10 + n], {n, 1, 60}] (* Vaclav Kotesovec, Jul 05 2025 *)

G.f.: Sum_{m>0} (x^(4*m) / Product_{i>m} (1-x^i)). More generally, g.f. for number of partitions of n such that the least part occurs exactly k times is Sum_{m>0} (x^(k*m) / Product_{i>m} (1-x^i)). Vladeta Jovovic
From Vaclav Kotesovec, Jul 05 2025: (Start)
a(n) = -p(n) + 4*p(n+4) - p(n+5) - 2*p(n+6) - 2*p(n+7) + p(n+8) + 2*p(n+9) - p(n+10), where p(n) = A000041(n).
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)) * (1 - (3^(3/2)/(Pi*sqrt(2)) + 61*Pi/(24*sqrt(6)))/sqrt(n)). (End)

A339887 Number of factorizations of n into primes or squarefree semiprimes.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 5, 1, 2, 2, 1, 2, 4, 1, 2, 2, 4, 1, 3, 1, 2, 2, 2, 2, 4, 1, 2, 1, 2, 1, 5, 2, 2, 2

Offset: 1

Gus Wiseman, Dec 22 2020

nonn

A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
Conjecture: also the number of semistandard Young tableaux whose entries are the prime indices of n (A323437).
Is this a duplicate of A323437? - R. J. Mathar, Jan 05 2021

			The a(n) factorizations for n = 36, 60, 180, 360, 420, 840:
  6*6       6*10      5*6*6       6*6*10        2*6*35      6*10*14
  2*3*6     2*5*6     2*6*15      2*5*6*6       5*6*14      2*2*6*35
  2*2*3*3   2*2*15    3*6*10      2*2*6*15      6*7*10      2*5*6*14
            2*3*10    2*3*5*6     2*3*6*10      2*10*21     2*6*7*10
            2*2*3*5   2*2*3*15    2*2*3*5*6     2*14*15     2*2*10*21
                      2*3*3*10    2*2*2*3*15    2*5*6*7     2*2*14*15
                      2*2*3*3*5   2*2*3*3*10    3*10*14     2*2*5*6*7
                                  2*2*2*3*3*5   2*2*3*35    2*3*10*14
                                                2*2*5*21    2*2*2*3*35
                                                2*2*7*15    2*2*2*5*21
                                                2*3*5*14    2*2*2*7*15
                                                2*3*7*10    2*2*3*5*14
                                                2*2*3*5*7   2*2*3*7*10
                                                            2*2*2*3*5*7

Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

See link for additional cross-references.
Only allowing only primes gives A008966.
Not allowing primes gives A320656.
Unlabeled multiset partitions of this type are counted by A320663/A339888.
Allowing squares of primes gives A320732.
The strict version is A339742.
A001055 counts factorizations.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
Cf. A000070, A000961, A001221, A096373, A320893, A338914, A339740, A339741, A339841, A339846.

sqpe[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqpe[n/d],Min@@#>=d&]],{d,Select[Divisors[n],PrimeQ[#]||SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
Table[Length[sqpe[n]],{n,100}]

a(A002110(n)) = A000085(n), and in general if n is a product of k distinct primes, a(n) = A000085(k).

a(n) = Sum_{d|n} A320656(n/d), so A320656 is the Moebius transform of this sequence.

A097093 Number of partitions of n such that the least part occurs exactly five times.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 2, 3, 4, 4, 8, 9, 14, 16, 23, 27, 39, 48, 62, 76, 100, 120, 159, 190, 241, 292, 367, 443, 552, 663, 816, 980, 1200, 1430, 1742, 2075, 2504, 2979, 3575, 4232, 5063, 5980, 7114, 8382, 9930, 11663, 13773, 16140, 18980, 22190, 26017

Offset: 1

Robert G. Wilson v, Jul 24 2004

nonn

In general, if k>=1 and g.f. = Sum_{m>0} (x^(k*m) / Product_{i>m} (1-x^i)), then a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)) * (1 - (3^(3/2)/(Pi*sqrt(2)) + (24*k - 35)*Pi/(24*sqrt(6)))/sqrt(n)). - Vaclav Kotesovec, Jul 05 2025

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A002865, A096373, A097091, A097092.

f[n_] := Block[{p = IntegerPartitions[n], l = PartitionsP[n], c = 0, k = 1}, While[k < l + 1, q = PadLeft[ p[[k]], 6]; If[ q[[1]] != q[[6]] && q[[2]] == q[[6]], c++ ]; k++ ]; c]; Table[ f[n], {n, 54}]
Table[Count[IntegerPartitions[n],?(Length[Split[#][[-1]]]==5&)],{n,60}] (* _Harvey P. Dale, Feb 07 2022 *)
nmax = 60; Rest[CoefficientList[Series[Sum[x^(5*m)/Product[1-x^k,{k,m+1,nmax}], {m, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 04 2025 *)
Table[-PartitionsP[n] + 5 PartitionsP[5 + n] - PartitionsP[6 + n] - 2 PartitionsP[7 + n] - 2 PartitionsP[8 + n] - 3 PartitionsP[9 + n] + 3 PartitionsP[10 + n] + PartitionsP[11 + n] + 2 PartitionsP[12 + n] - PartitionsP[13 + n] - 2 PartitionsP[14 + n] + PartitionsP[15 + n], {n, 1, 60}] (* Vaclav Kotesovec, Jul 05 2025 *)

G.f.: Sum_{m>0} (x^(5*m) / Product_{i>m} (1-x^i)). More generally, g.f. for number of partitions of n such that the least part occurs exactly k times is Sum_{m>0} (x^(k*m) / Product_{i>m} (1-x^i)). Vladeta Jovovic
From Vaclav Kotesovec, Jul 05 2025: (Start)
a(n) = -p(n) + 5*p(n+5) - p(n+6) - 2*p(n+7) - 2*p(n+8) - 3*p(n+9) + 3*p(n+10) + p(n+11) + 2*p(n+12) - p(n+13) - 2*p(n+14) + p(n+15), where p(n) = A000041(n).
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)) * (1 - (3^(3/2)/(Pi*sqrt(2)) + 85*Pi/(24*sqrt(6)))/sqrt(n)). (End)

A339842 Heinz numbers of non-graphical, multigraphical integer partitions of even numbers.

Original entry on oeis.org

9, 25, 30, 49, 63, 70, 75, 84, 100, 121, 147, 154, 165, 169, 175, 189, 196, 198, 210, 220, 250, 264, 273, 280, 286, 289, 325, 343, 351, 361, 363, 364, 385, 390, 441, 442, 462, 468, 484, 490, 495, 507, 520, 525, 529, 550, 561, 588, 594, 595, 616, 624, 637, 646

Offset: 1

Gus Wiseman, Dec 27 2020

nonn

An integer partition is graphical if it comprises the multiset of vertex-degrees of some graph, and multigraphical if it comprises the multiset of vertex-degrees of some multigraph.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      9: {2,2}        189: {2,2,2,4}      363: {2,5,5}
     25: {3,3}        196: {1,1,4,4}      364: {1,1,4,6}
     30: {1,2,3}      198: {1,2,2,5}      385: {3,4,5}
     49: {4,4}        210: {1,2,3,4}      390: {1,2,3,6}
     63: {2,2,4}      220: {1,1,3,5}      441: {2,2,4,4}
     70: {1,3,4}      250: {1,3,3,3}      442: {1,6,7}
     75: {2,3,3}      264: {1,1,1,2,5}    462: {1,2,4,5}
     84: {1,1,2,4}    273: {2,4,6}        468: {1,1,2,2,6}
    100: {1,1,3,3}    280: {1,1,1,3,4}    484: {1,1,5,5}
    121: {5,5}        286: {1,5,6}        490: {1,3,4,4}
    147: {2,4,4}      289: {7,7}          495: {2,2,3,5}
    154: {1,4,5}      325: {3,3,6}        507: {2,6,6}
    165: {2,3,5}      343: {4,4,4}        520: {1,1,1,3,6}
    169: {6,6}        351: {2,2,2,6}      525: {2,3,3,4}
    175: {3,3,4}      361: {8,8}          529: {9,9}
For example, a complete list of all multigraphs with degrees (4,2,2,2) is:
  {{1,2},{1,2},{1,3},{1,4},{3,4}}
  {{1,2},{1,3},{1,3},{1,4},{2,4}}
  {{1,2},{1,3},{1,4},{1,4},{2,3}}
Since none of these is strict, i.e., a graph, the Heinz number 189 is in the sequence.

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

See link for additional cross references.
Distinct prime shadows (images under A181819) of A340017.
A000070 counts non-multigraphical partitions (A339620).
A000569 counts graphical partitions (A320922).
A027187 counts partitions of even length (A028260).
A058696 counts partitions of even numbers (A300061).
A096373 cannot be partitioned into strict pairs.
A209816 counts multigraphical partitions (A320924).
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A320893 can be partitioned into distinct pairs but not into strict pairs.
A339560 can be partitioned into distinct strict pairs.
A339617 counts non-graphical partitions of 2n (A339618).
A339659 counts graphical partitions of 2n into k parts.
Cf. A004251, A006129, A007717, A056239, A095268, A112798, A181821, A305936, A318284, A339559.

strr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strr[n/d],Min@@#>=d&]],{d,Select[Divisors[n],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],EvenQ[Length[nrmptn[#]]]&& Select[strr[Times@@Prime/@nrmptn[#]],UnsameQ@@#&]=={}&&strr[Times@@Prime/@nrmptn[#]]!={}&]

Equals A320924 /\ A339618.

Equals A320924 \ A320922.

A096144 Triangle T(n,k) = number of partitions of n in which the least part occurs exactly k times, k=1..n.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 2, 2, 0, 1, 4, 1, 1, 0, 1, 4, 3, 2, 1, 0, 1, 7, 3, 2, 1, 1, 0, 1, 8, 6, 2, 3, 1, 1, 0, 1, 12, 5, 6, 2, 2, 1, 1, 0, 1, 14, 11, 5, 4, 3, 2, 1, 1, 0, 1, 21, 11, 8, 5, 4, 2, 2, 1, 1, 0, 1, 24, 17, 11, 9, 4, 5, 2, 2, 1, 1, 0, 1, 34, 20, 15, 9, 8, 4, 4, 2, 2, 1, 1, 0, 1, 41, 30, 18, 14, 9, 7, 5, 4, 2, 2, 1, 1, 0, 1

Offset: 1

Vladeta Jovovic, Jul 24 2004

Reversed rows converge to A002865. - Joerg Arndt, Jul 07 2014

T(n,k) is the number of partitions of n for which the difference between the two largest distinct parts is k (in partitions having only 1 part, we assume that 0 is also a part). This follows easily from the definition by taking the conjugate partitions. Example: T(7,2) = 3 because we have [3,1,1,1,1], [3,3,1], and [4,2,1]. - Emeric Deutsch, Dec 05 2015.

			Triangle starts:
01:  1
02:  1 1
03:  2 0 1
04:  2 2 0 1
05:  4 1 1 0 1
06:  4 3 2 1 0 1
07:  7 3 2 1 1 0 1
08:  8 6 2 3 1 1 0 1
09: 12 5 6 2 2 1 1 0 1
10: 14 11 5 4 3 2 1 1 0 1
11: 21 11 8 5 4 2 2 1 1 0 1
12: 24 17 11 9 4 5 2 2 1 1 0 1
13: 34 20 15 9 8 4 4 2 2 1 1 0 1
14: 41 30 18 14 9 7 5 ...
T(7,2)=3 because we have: 5+1+1, 3+2+2, 3+2+1+1. - _Geoffrey Critzer_, Jun 20 2014

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A002865 (first column), A096373 (second column), A000041 (row sums).

T(2n,n) gives A232697(n). - Alois P. Heinz, Jun 20 2014

b:= proc(n, i) option remember; `if`(i=1, x^n,
      `if`(irem(n, i, 'k')=0, x^k, 0)+
       add(b(n-i*j, i-1), j=0..(n-1)/i))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
seq(T(n), n=1..20);  # Alois P. Heinz, Jun 20 2014

nn=20;Table[Take[Map[Drop[#,1]&,Drop[CoefficientList[Series[Sum[y x^k/(1-y x^k) Product[1/(1- x^j),{j,k+1,nn}],{k,1,nn}],{x,0,nn}],{x,y}],1]][[i]],i],{i,1,nn}]//Grid (* Geoffrey Critzer, Jun 20 2014 *)

G.f. for k-th column: sum(m>=1, x^(k*m)/prod(i>=m+1, 1-x^i ) ).

A238495 Number of partitions p of n such that min(p) + (number of parts of p) is not a part of p.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 14, 19, 27, 36, 51, 66, 90, 118, 156, 201, 264, 336, 434, 550, 700, 880, 1112, 1385, 1733, 2149, 2666, 3283, 4049, 4956, 6072, 7398, 9009, 10922, 13237, 15970, 19261, 23147, 27790, 33260, 39776, 47425, 56497, 67133, 79685, 94371, 111653

Offset: 1

Clark Kimberling, Feb 27 2014

Also the number of integer partitions of n + 1 with median > 1, or with no more 1's than non-1 parts. - Gus Wiseman, Jul 10 2023

			a(6) = 9 counts all the 11 partitions of 6 except 42 and 411.
From _Gus Wiseman_, Jul 10 2023 (Start)
The a(2) = 1 through a(8) = 14 partitions:
  (2)  (3)   (4)   (5)    (6)     (7)     (8)
       (21)  (22)  (32)   (33)    (43)    (44)
             (31)  (41)   (42)    (52)    (53)
                   (221)  (51)    (61)    (62)
                          (222)   (322)   (71)
                          (321)   (331)   (332)
                          (2211)  (421)   (422)
                                  (2221)  (431)
                                  (3211)  (521)
                                          (2222)
                                          (3221)
                                          (3311)
                                          (4211)
                                          (22211)
(End)

Cf. A096373.
For mean instead of median we have A000065, ranks A057716.
The complement is counted by A027336, ranks A364056.
Rows sums of A359893 if we remove the first column.
These partitions have ranks A364058.
A000041 counts integer partitions.
A008284 counts partitions by length, A058398 by mean.
A025065 counts partitions with low mean 1, ranks A363949.
A124943 counts partitions by low median, high A124944.
A241131 counts partitions with low mode 1, ranks A360015.
Cf. A000070, A000975, A002865, A110618, A237984, A363488.

Table[Count[IntegerPartitions[n], p_ /; ! MemberQ[p, Length[p] + Min[p]]], {n, 50}]
Table[Length[Select[IntegerPartitions[n+1],Median[#]>1&]],{n,30}] (* Gus Wiseman, Jul 10 2023 *)

From Gus Wiseman, Jul 11 2023: (Start)
a(n>2) = A000041(n) - A096373(n-2).
a(n>1) = A000041(n-2) + A002865(n+1).
a(n) = A000041(n+1) - A027336(n).
(End)

Formula corrected by Gus Wiseman, Jul 11 2023

cp's OEIS Frontend

A339740 Non-products of distinct primes or squarefree semiprimes.

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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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A339843 Number of distinct sorted degree sequences among all n-vertex half-loop-graphs without isolated vertices.

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A097091 Number of partitions of n such that the least part occurs exactly three times.

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A097092 Number of partitions of n such that the least part occurs exactly four times.

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A339887 Number of factorizations of n into primes or squarefree semiprimes.

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A097093 Number of partitions of n such that the least part occurs exactly five times.

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A339842 Heinz numbers of non-graphical, multigraphical integer partitions of even numbers.

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