A320891 -id:A320891 - cp's OEIS frontend

A339559 Number of integer partitions of n that have an even number of parts and cannot be partitioned into distinct pairs of distinct parts, i.e., that are not the multiset union of any set of edges.

0, 0, 1, 0, 2, 1, 4, 3, 7, 6, 14, 14, 23, 27, 41, 47, 70, 84, 114, 141, 190, 225, 303, 370, 475, 578, 738, 890, 1131, 1368, 1698, 2058, 2549, 3048, 3759, 4505, 5495, 6574, 7966, 9483, 11450, 13606, 16307, 19351, 23116, 27297, 32470, 38293, 45346, 53342, 62939

Offset: 0

Gus Wiseman, Dec 10 2020

nonn

The multiplicities of such a partition form a non-graphical partition.

			The a(2) = 1 through a(10) = 14 partitions (empty column indicated by dot):
  11   .   22     2111   33       2221     44         3222       55
           1111          2211     4111     2222       6111       3322
                         3111     211111   3311       222111     3331
                         111111            5111       321111     4222
                                           221111     411111     4411
                                           311111     21111111   7111
                                           11111111              222211
                                                                 322111
                                                                 331111
                                                                 421111
                                                                 511111
                                                                 22111111
                                                                 31111111
                                                                 1111111111
For example, the partition y = (4,4,3,3,2,2,1,1,1,1) can be partitioned into a multiset of edges in just three ways:
  {{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}}
None of these are strict, so y is counted under a(22).

Eric Weisstein's World of Mathematics, Graphical partition.

A320894 ranks these partitions (using Heinz numbers).
A338915 allows equal pairs (x,x).
A339560 counts the complement in even-length partitions.
A339564 counts factorizations of the same type.
A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
A000569 counts graphical partitions, ranked by A320922.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A058696 counts partitions of even numbers, ranked by A300061.
A209816 counts multigraphical partitions, ranked by A320924.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339617 counts non-graphical partitions of 2n, ranked by A339618.
A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001055, A001221, A005117, A007717, A025065, A030229, A089259, A292432, A320893, A338899, A338903, A339619.

strs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&strs[Times@@Prime/@#]=={}&]],{n,0,15}]

A027187(n) = a(n) + A339560(n).

More terms from Jinyuan Wang, Feb 14 2025

A339657 Heinz numbers of non-loop-graphical partitions of even numbers.

Original entry on oeis.org

7, 13, 19, 21, 22, 29, 34, 37, 39, 43, 46, 49, 52, 53, 55, 57, 61, 62, 66, 71, 76, 79, 82, 85, 87, 89, 91, 94, 101, 102, 107, 111, 113, 115, 116, 117, 118, 121, 129, 130, 131, 133, 134, 136, 138, 139, 146, 148, 151, 154, 155, 156, 159, 163, 165, 166, 169, 171

Offset: 1

Gus Wiseman, Dec 18 2020

nonn

Equals the image of A181819 applied to the set of terms of A320892.
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.
An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with two equal vertices. Loop-graphical partitions are counted by A339656, with Heinz numbers A339658.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct pairs, i.e., into a set of edges and loops;
(2) n can be factored into distinct semiprimes;
(3) the prime signature of n is loop-graphical.

			The sequence of terms together with their prime indices begins:
      7: {4}         57: {2,8}      107: {28}
     13: {6}         61: {18}       111: {2,12}
     19: {8}         62: {1,11}     113: {30}
     21: {2,4}       66: {1,2,5}    115: {3,9}
     22: {1,5}       71: {20}       116: {1,1,10}
     29: {10}        76: {1,1,8}    117: {2,2,6}
     34: {1,7}       79: {22}       118: {1,17}
     37: {12}        82: {1,13}     121: {5,5}
     39: {2,6}       85: {3,7}      129: {2,14}
     43: {14}        87: {2,10}     130: {1,3,6}
     46: {1,9}       89: {24}       131: {32}
     49: {4,4}       91: {4,6}      133: {4,8}
     52: {1,1,6}     94: {1,15}     134: {1,19}
     53: {16}       101: {26}       136: {1,1,1,7}
     55: {3,5}      102: {1,2,7}    138: {1,2,9}
For example, the three loop-multigraphs with degrees y = (5,2,1) are:
  {{1,1},{1,1},{1,2},{2,3}}
  {{1,1},{1,1},{1,3},{2,2}}
  {{1,1},{1,2},{1,2},{1,3}},
but since none of these is a loop-graph (they have multiple edges), the Heinz number 66 is in the sequence.

Eric Weisstein's World of Mathematics, Graphical partition.

A320892 has these prime shadows (see A181819).
A321728 is conjectured to be the version for half-loops {x} instead of loops {x,x}.
A339655 counts these partitions.
A339658 ranks the complement, counted by A339656.
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A006881 lists squarefree semiprimes, with odd and even terms A046388 and A100484.
A101048 counts partitions into semiprimes.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339844 counts loop-graphical partitions by length.
factorizations of n into distinct primes or squarefree 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).
- A209816 counts multigraphical partitions (A320924).
- A339655 counts non-loop-graphical partitions of 2n (A339657 [this sequence]).
- A339656 counts loop-graphical partitions (A339658).
- A339617 counts non-graphical partitions of 2n (A339618).
- A000569 counts graphical partitions (A320922).
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001055, A001221, A001222, A007717, A056239, A112798, A320732, A338898, A338912, A338913, A339742, A339839.

spsbin[{}]:={{}};spsbin[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsbin[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@spsbin[Range[Length[set]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[50],EvenQ[Length[nrmptn[#]]]&&Select[mpsbin[nrmptn[#]],UnsameQ@@#&]=={}&]

A300061 = A339657 \/ A339658.

A339620 Heinz numbers of non-multigraphical partitions of even numbers.

Original entry on oeis.org

3, 7, 10, 13, 19, 21, 22, 28, 29, 34, 37, 39, 43, 46, 52, 53, 55, 57, 61, 62, 66, 71, 76, 79, 82, 85, 87, 88, 89, 91, 94, 101, 102, 107, 111, 113, 115, 116, 117, 118, 129, 130, 131, 133, 134, 136, 138, 139, 146, 148, 151, 155, 156, 159, 163, 166, 171, 172, 173

Offset: 1

Gus Wiseman, Dec 18 2020

nonn

An integer partition is non-multigraphical if it does not comprise the multiset of vertex-degrees of any multigraph (multiset of non-loop edges). Multigraphical partitions are counted by A209816, non-multigraphical partitions by A000070.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
The following are equivalent characteristics for any positive integer n:
(1) the multiset of prime indices of n can be partitioned into strict pairs (a multiset of edges);
(2) n can be factored into squarefree semiprimes;
(3) the unordered prime signature of n is multigraphical.

			The sequence of terms together with their prime indices begins:
      3: {2}         53: {16}          94: {1,15}
      7: {4}         55: {3,5}        101: {26}
     10: {1,3}       57: {2,8}        102: {1,2,7}
     13: {6}         61: {18}         107: {28}
     19: {8}         62: {1,11}       111: {2,12}
     21: {2,4}       66: {1,2,5}      113: {30}
     22: {1,5}       71: {20}         115: {3,9}
     28: {1,1,4}     76: {1,1,8}      116: {1,1,10}
     29: {10}        79: {22}         117: {2,2,6}
     34: {1,7}       82: {1,13}       118: {1,17}
     37: {12}        85: {3,7}        129: {2,14}
     39: {2,6}       87: {2,10}       130: {1,3,6}
     43: {14}        88: {1,1,1,5}    131: {32}
     46: {1,9}       89: {24}         133: {4,8}
     52: {1,1,6}     91: {4,6}        134: {1,19}
For example, a complete lists of all loop-multigraphs with degrees (5,2,1) is:
  {{1,1},{1,1},{1,2},{2,3}}
  {{1,1},{1,1},{1,3},{2,2}}
  {{1,1},{1,2},{1,2},{1,3}},
but since none of these is a multigraph (they have loops), the Heinz number 66 belongs to the sequence.

Eric Weisstein's World of Mathematics, Graphical partition.

A000070 counts these partitions.
A300061 is a superset.
A320891 has image under A181819 equal to this set of terms.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A320656 counts factorizations into squarefree 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 [this sequence]).
- A209816 counts multigraphical partitions (A320924).
- A147878 counts connected multigraphical partitions (A320925).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339656 counts loop-graphical partitions (A339658).
- A339617 counts non-graphical partitions of 2n (A339618).
- A000569 counts graphical partitions (A320922).
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001055, A005117, A007717, A030229, A050320, A056239, A112798, A320655, A338899, A339113, A339661.

prpts[m_]:=If[Length[m]==0,{{}},Join@@Table[Prepend[#,ipr]&/@prpts[Fold[DeleteCases[#1,#2,{1},1]&,m,ipr]],{ipr,Select[Subsets[Union[m],{2}],MemberQ[#,m[[1]]]&]}]];
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[#]]]&&prpts[nrmptn[#]]=={}&]

Equals A300061 \ A320924.

For all n, both A181821(a(n)) and A304660(a(n)) belong to A320891.

A339661 Number of factorizations of n into distinct squarefree semiprimes.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0

Offset: 1

Gus Wiseman, Dec 19 2020

nonn

A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

Also the number of strict multiset partitions of the multiset of prime factors of n, into distinct strict pairs.

			The a(n) factorizations for n = 210, 1260, 4620, 30030, 69300 are respectively 3, 2, 6, 15, 7:
  (6*35)   (6*10*21)  (6*10*77)   (6*55*91)    (6*10*15*77)
  (10*21)  (6*14*15)  (6*14*55)   (6*65*77)    (6*10*21*55)
  (14*15)             (6*22*35)   (10*33*91)   (6*10*33*35)
                      (10*14*33)  (10*39*77)   (6*14*15*55)
                      (10*21*22)  (14*33*65)   (6*15*22*35)
                      (14*15*22)  (14*39*55)   (10*14*15*33)
                                  (15*22*91)   (10*15*21*22)
                                  (15*26*77)
                                  (21*22*65)
                                  (21*26*55)
                                  (22*35*39)
                                  (26*33*35)
                                  (6*35*143)
                                  (10*21*143)
                                  (14*15*143)

Antti Karttunen, Table of n, a(n) for n = 1..69300
Index entries for sequences computed from exponents in factorization of n

Dirichlet convolution of A008836 (Liouville's lambda) with A339742.
A050326 allows all squarefree numbers, non-strict case A050320.
A320656 is the not necessarily strict version.
A320911 lists all (not just distinct) products of squarefree semiprimes.
A322794 counts uniform factorizations, such as these.
A339561 lists positions of nonzero terms.
A001055 counts factorizations, with strict case A045778.
A001358 lists semiprimes, with squarefree case A006881.
A320655 counts factorizations into semiprimes, with strict case A322353.
The following count vertex-degree partitions and give their Heinz numbers:
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A209816 counts multigraphical partitions (A320924).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339656 counts loop-graphical partitions (A339658).
- A339617 counts non-graphical partitions of 2n (A339618).
- A000569 counts graphical partitions (A320922).
The following count partitions of even length and give their Heinz numbers:
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001221, A005117, A007716, A028260, A280710, A300061, A320658, A320659, A320923, A330974.

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

A280710(n) = (bigomega(n)==2*issquarefree(n)); \\ From A280710.
A339661(n, u=(1+n)) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (dA280710(d), s += A339661(n/d, d))); (s)); \\ Antti Karttunen, May 02 2022

a(n) = Sum_{d|n} (-1)^A001222(d) * A339742(n/d).

More terms and secondary offset added by Antti Karttunen, May 02 2022

A339740 Non-products of distinct primes or squarefree semiprimes.

Original entry on oeis.org

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[#]=={}&]

A339191 Partial products of squarefree semiprimes (A006881).

Original entry on oeis.org

6, 60, 840, 12600, 264600, 5821200, 151351200, 4994589600, 169816046400, 5943561624000, 225855341712000, 8808358326768000, 405184483031328000, 20664408634597728000, 1136542474902875040000, 64782921069463877280000, 3757409422028904882240000

Offset: 1

Gus Wiseman, Nov 30 2020

nonn

A squarefree semiprime is a product of any two distinct prime numbers.

Do all terms belong to A242031 (weakly decreasing prime signature)?

			The sequence of terms together with their prime indices begins:
          6: {1,2}
         60: {1,1,2,3}
        840: {1,1,1,2,3,4}
      12600: {1,1,1,2,2,3,3,4}
     264600: {1,1,1,2,2,2,3,3,4,4}
    5821200: {1,1,1,1,2,2,2,3,3,4,4,5}
  151351200: {1,1,1,1,1,2,2,2,3,3,4,4,5,6}
The sequence of terms together with their prime signatures begins:
                   6: (1,1)
                  60: (2,1,1)
                 840: (3,1,1,1)
               12600: (3,2,2,1)
              264600: (3,3,2,2)
             5821200: (4,3,2,2,1)
           151351200: (5,3,2,2,1,1)
          4994589600: (5,4,2,2,2,1)
        169816046400: (6,4,2,2,2,1,1)
       5943561624000: (6,4,3,3,2,1,1)
     225855341712000: (7,4,3,3,2,1,1,1)
    8808358326768000: (7,5,3,3,2,2,1,1)
  405184483031328000: (8,5,3,3,2,2,1,1,1)

A000040 lists the primes, with partial products A002110 (primorials).
A001358 lists semiprimes, with partial products A112141.
A002100 counts partitions into squarefree semiprimes (restricted: A338903)
A000142 lists factorial numbers, with partial products A000178.
A005117 lists squarefree numbers, with partial products A111059.
A006881 lists squarefree semiprimes, with partial sums A168472.
A166237 gives first differences of squarefree semiprimes.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A338898/A338912/A338913 give prime indices of semiprimes.
A338899/A270650/A270652 give prime indices of squarefree semiprimes.
A338901 gives first appearances in the list of squarefree semiprimes.
A339113 gives products of primes of squarefree semiprime index.
Cf. A001221, A112798, A167171, A320732, A320891, A320892, A320894, A320911, A338900, A338902, A339003, A339004.

FoldList[Times,Select[Range[20],SquareFreeQ[#]&&PrimeOmega[#]==2&]]

A340017 Products of squarefree semiprimes that are not products of distinct squarefree semiprimes.

Original entry on oeis.org

36, 100, 196, 216, 225, 360, 441, 484, 504, 540, 600, 676, 756, 792, 936, 1000, 1089, 1156, 1176, 1188, 1224, 1225, 1296, 1350, 1368, 1400, 1404, 1444, 1500, 1521, 1656, 1836, 1960, 2052, 2088, 2116, 2160, 2200, 2232, 2250, 2484, 2600, 2601, 2646, 2664, 2744

Offset: 1

Gus Wiseman, Dec 30 2020

nonn

Of course, every number is a product of squarefree numbers (A050320).
A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
All terms have even Omega (A001222, A028260).

			The sequence of terms together with their prime indices begins:
      36: {1,1,2,2}        1000: {1,1,1,3,3,3}
     100: {1,1,3,3}        1089: {2,2,5,5}
     196: {1,1,4,4}        1156: {1,1,7,7}
     216: {1,1,1,2,2,2}    1176: {1,1,1,2,4,4}
     225: {2,2,3,3}        1188: {1,1,2,2,2,5}
     360: {1,1,1,2,2,3}    1224: {1,1,1,2,2,7}
     441: {2,2,4,4}        1225: {3,3,4,4}
     484: {1,1,5,5}        1296: {1,1,1,1,2,2,2,2}
     504: {1,1,1,2,2,4}    1350: {1,2,2,2,3,3}
     540: {1,1,2,2,2,3}    1368: {1,1,1,2,2,8}
     600: {1,1,1,2,3,3}    1400: {1,1,1,3,3,4}
     676: {1,1,6,6}        1404: {1,1,2,2,2,6}
     756: {1,1,2,2,2,4}    1444: {1,1,8,8}
     792: {1,1,1,2,2,5}    1500: {1,1,2,3,3,3}
     936: {1,1,1,2,2,6}    1521: {2,2,6,6}
For example, a complete list of all factorizations of 7560 into squarefree semiprimes is:
  7560 = (6*6*6*35) = (6*6*10*21) = (6*6*14*15),
but since none of these is strict, 7560 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.
The distinct prime shadows (under A181819) of these terms are A339842.
Factorizations into squarefree semiprimes are counted by A320656.
Products of squarefree semiprimes that are not products of distinct semiprimes are A320893.
Factorizations into distinct squarefree semiprimes are A339661.
For the next four lines, we list numbers with even Omega (A028260).
- A320891 cannot be factored into squarefree semiprimes.
- A320894 cannot be factored into distinct squarefree semiprimes.
- A320911 can be factored into squarefree semiprimes.
- A339561 can be factored into distinct squarefree semiprimes.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A030229 lists squarefree numbers with even Omega.
A050320 counts factorizations into squarefree numbers.
A050326 counts factorizations into distinct squarefree numbers.
A181819 is the Heinz number of the prime signature of n (prime shadow).
A320656 counts factorizations into squarefree semiprimes.
A339560 can be partitioned into distinct strict pairs.
Cf. A001055, A001222, A005117, A007717, A096373, A112798, A300061, A322353, A339559, A338899, A339740.

strr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strr[n/d],Min@@#>=d&]],{d,Select[Divisors[n],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
Select[Range[1000],Select[strr[#],UnsameQ@@#&]=={}&&strr[#]!={}&]

Equals A320894 /\ A320911.

Numbers n such that A320656(n) > 0 but A339661(n) = 0.

cp's OEIS Frontend

A339559 Number of integer partitions of n that have an even number of parts and cannot be partitioned into distinct pairs of distinct parts, i.e., that are not the multiset union of any set of edges.

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A339657 Heinz numbers of non-loop-graphical partitions of even numbers.

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A339620 Heinz numbers of non-multigraphical partitions of even numbers.

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A339661 Number of factorizations of n into distinct squarefree semiprimes.

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A339740 Non-products of distinct primes or squarefree semiprimes.

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A339191 Partial products of squarefree semiprimes (A006881).

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A340017 Products of squarefree semiprimes that are not products of distinct squarefree semiprimes.

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