A339839 -id:A339839 - cp's OEIS frontend

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

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.

A322353 Number of factorizations of n into distinct semiprimes; a(1) = 1 by convention.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 2, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 2, 1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0

Offset: 1

Antti Karttunen, Dec 06 2018

nonn

A semiprime (A001358) is a product of any two prime numbers. In the even case, these factorizations have A001222(n)/2 factors. - Gus Wiseman, Dec 31 2020

Records 1, 2, 3, 4, 5, 9, 13, 15, 17, ... occur at 1, 60, 210, 840, 1260, 4620, 27720, 30030, 69300, ...

			a(4) = 1, as there is just one way to factor 4 into distinct semiprimes, namely as {4}.
From _Gus Wiseman_, Dec 31 2020: (Start)
The a(n) factorizations for n = 60, 210, 840, 1260, 4620, 12600, 18480:
  4*15   6*35    4*6*35    4*9*35    4*15*77    4*6*15*35    4*6*10*77
  6*10   10*21   4*10*21   4*15*21   4*21*55    4*6*21*25    4*6*14*55
         14*15   4*14*15   6*10*21   4*33*35    4*9*10*35    4*6*22*35
                 6*10*14   6*14*15   6*10*77    4*9*14*25    4*10*14*33
                           9*10*14   6*14*55    4*10*15*21   4*10*21*22
                                     6*22*35    6*10*14*15   4*14*15*22
                                     10*14*33                6*10*14*22
                                     10*21*22
                                     14*15*22
(End)

Unlabeled multiset partitions of this type are counted by A007717.
The version for partitions is A112020, or A101048 without distinctness.
The non-strict version is A320655.
Positions of zeros include A320892.
Positions of nonzero terms are A320912.
The case of squarefree factors is A339661, or A320656 without distinctness.
Allowing prime factors gives A339839, or A320732 without distinctness.
A322661 counts loop-graphs, ranked by A320461.
A001055 counts factorizations, with strict case A045778.
A001358 lists semiprimes, with squarefree case A006881.
A027187 counts partitions of even length, ranked by A028260.
A037143 lists primes and semiprimes.
A338898/A338912/A338913 give the prime indices of semiprimes.
A339846 counts even-length factorizations, with ordered version A174725.
Cf. A001221, A006125, A006129, A028260, A320893, A338915, A339841.

Table[Count[Subsets[Select[Divisors[n], PrimeOmega[#] == 2 &]], ?(Times @@ # == n &)], {n, 105}] (* _Michael De Vlieger, Dec 11 2020 *)

A322353(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((2==bigomega(d)&&(d<=m)), s += A322353(n/d, d-1))); (s)); \\ Antti Karttunen, Dec 10 2020

a(n) = Sum_{d|n} (-1)^A001222(d) * A339839(n/d). - Gus Wiseman, Dec 31 2020

A339742 Number of factorizations of n into distinct primes or squarefree semiprimes.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 4, 1, 0, 2, 2, 2, 1, 1, 2, 2, 0, 1, 4, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 3, 1, 2, 1, 0, 2, 4, 1, 1, 2, 4, 1, 0, 1, 2, 1, 1, 2, 4, 1, 0, 0, 2, 1, 3, 2, 2, 2, 0, 1, 3, 2, 1, 2, 2, 2, 0, 1, 1, 1, 1, 1, 4, 1, 0, 4

Offset: 1

Gus Wiseman, Dec 20 2020

nonn

A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct singletons or strict pairs, i.e., into a set of half-loops and edges;
(2) n can be factored into distinct primes or squarefree semiprimes.

			The a(n) factorizations for n = 6, 30, 60, 210, 420 are respectively 2, 4, 3, 10, 9:
  (6)    (5*6)    (6*10)    (6*35)     (2*6*35)
  (2*3)  (2*15)   (2*5*6)   (10*21)    (5*6*14)
         (3*10)   (2*3*10)  (14*15)    (6*7*10)
         (2*3*5)            (5*6*7)    (2*10*21)
                            (2*3*35)   (2*14*15)
                            (2*5*21)   (2*5*6*7)
                            (2*7*15)   (3*10*14)
                            (3*5*14)   (2*3*5*14)
                            (3*7*10)   (2*3*7*10)
                            (2*3*5*7)

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 A008966 with A339661.
A008966 allows only primes.
A339661 does not allow primes, only squarefree semiprimes.
A339740 lists the positions of zeros.
A339741 lists the positions of positive terms.
A339839 allows nonsquarefree semiprimes.
A339887 is the non-strict version.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A013929 cannot be factored into distinct primes.
A293511 are a product of distinct squarefree numbers in exactly one way.
A320663 counts non-isomorphic multiset partitions into singletons or pairs.
A339840 cannot be factored into distinct primes or semiprimes.
A339841 have exactly one factorization into primes or semiprimes.
The following count factorizations:
- A001055 into all positive integers > 1.
- A050320 into squarefree numbers.
- A050326 into distinct squarefree numbers.
- A320655 into semiprimes.
- A320656 into squarefree semiprimes.
- A320732 into primes or semiprimes.
- A322353 into distinct semiprimes.
- A339742 [this sequence] into distinct primes or squarefree semiprimes.
- A339839 into distinct primes or semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A000569 counts graphical partitions (A320922).
- A058696 counts all partitions of 2n (A300061).
- A209816 counts multigraphical partitions (A320924).
- A339656 counts loop-graphical partitions (A339658).
-
The following count partitions/factorizations of even length and give their Heinz numbers:
- A027187/A339846 has no additional conditions (A028260).
- A338914/A339562 can be partitioned into edges (A320911).
- A338916/A339563 can be partitioned into distinct pairs (A320912).
- A339559/A339564 cannot be partitioned into distinct edges (A320894).
- A339560/A339619 can be partitioned into distinct edges (A339561).
Cf. A000070, A001221, A005117, A320893, A320923, A338899, A339113, A339617, A353471.

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

A353471(n) = (numdiv(n)==2*omega(n));
A339742(n, u=(1+n)) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (dA353471(d), s += A339742(n/d, d))); (s)); \\ Antti Karttunen, May 02 2022

a(n) = Sum_{d|n squarefree} A339661(n/d).

More terms from Antti Karttunen, May 02 2022

A339841 Numbers that can be factored into distinct primes or semiprimes in exactly one way.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 48, 49, 53, 59, 61, 67, 71, 73, 79, 80, 83, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 131, 137, 139, 144, 149, 151, 157, 162, 163, 167, 169, 173, 176, 179, 181, 191, 193

Offset: 1

Gus Wiseman, Dec 25 2020

nonn

A semiprime (A001358) is a product of any two prime numbers.

			The sequence of terms together with their one factorization begins:
     1 =        29 = 29        80 = 2*4*10
     2 = 2      31 = 31        83 = 83
     3 = 3      37 = 37        89 = 89
     4 = 4      41 = 41        97 = 97
     5 = 5      43 = 43       101 = 101
     7 = 7      47 = 47       103 = 103
     8 = 2*4    48 = 2*4*6    107 = 107
     9 = 9      49 = 49       109 = 109
    11 = 11     53 = 53       112 = 2*4*14
    13 = 13     59 = 59       113 = 113
    17 = 17     61 = 61       121 = 121
    19 = 19     67 = 67       125 = 5*25
    23 = 23     71 = 71       127 = 127
    25 = 25     73 = 73       131 = 131
    27 = 3*9    79 = 79       137 = 137
For example, we have 360 = 2*3*6*10, so 360 is in the sequence. But 360 is absent from A293511, because we also have 360 = 2*6*30.

Amiram Eldar, Table of n, a(n) for n = 1..10000
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.
These are the positions of ones in A339839.
The version for no factorizations is A339840.
The version for at least one factorization is A339889.
A001055 counts factorizations.
A001358 lists semiprimes, with squarefree case A006881.
A037143 lists primes and semiprimes.
A293511 are a product of distinct squarefree numbers in exactly one way.
A320663 counts non-isomorphic multiset partitions into singletons or pairs.
A338915 counts partitions that cannot be partitioned into distinct pairs.
Cf. A002494, A013929, A028260, A320893, A320922, A339618, A339740, A339846.

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

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[#]=={}&]

A339840 Numbers that cannot be factored into distinct primes or semiprimes.

Original entry on oeis.org

16, 32, 64, 81, 96, 128, 160, 192, 224, 243, 256, 288, 320, 352, 384, 416, 448, 486, 512, 544, 576, 608, 625, 640, 704, 729, 736, 768, 800, 832, 864, 896, 928, 960, 972, 992, 1024, 1088, 1152, 1184, 1215, 1216, 1280, 1312, 1344, 1376, 1408, 1458, 1472, 1504

Offset: 1

Gus Wiseman, Dec 20 2020

nonn

A semiprime (A001358) is a product of any two prime numbers.

			The sequence of terms together with their prime indices begins:
    16: {1,1,1,1}
    32: {1,1,1,1,1}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
    96: {1,1,1,1,1,2}
   128: {1,1,1,1,1,1,1}
   160: {1,1,1,1,1,3}
   192: {1,1,1,1,1,1,2}
   224: {1,1,1,1,1,4}
   243: {2,2,2,2,2}
   256: {1,1,1,1,1,1,1,1}
   288: {1,1,1,1,1,2,2}
   320: {1,1,1,1,1,1,3}
   352: {1,1,1,1,1,5}
   384: {1,1,1,1,1,1,1,2}
   416: {1,1,1,1,1,6}
   448: {1,1,1,1,1,1,4}
   486: {1,2,2,2,2,2}
For example, a complete list of all factorizations of 192 into primes or semiprimes is:
  (2*2*2*2*2*2*3)
  (2*2*2*2*2*6)
  (2*2*2*2*3*4)
  (2*2*2*4*6)
  (2*2*3*4*4)
  (2*4*4*6)
  (3*4*4*4)
Since none of these is strict, 192 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Allowing only primes gives A013929.
Removing all squares of primes gives A339740.
These are the positions of zeros in A339839.
The complement is A339889.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A293511 are a product of distinct squarefree numbers in exactly one way.
A320663 counts non-isomorphic multiset partitions into singletons or pairs.
A338915 cannot be partitioned into distinct pairs (A320892).
A339841 have exactly one factorization into primes or semiprimes.
The following count factorizations:
- A001055 into all positive integers > 1.
- A320655 into semiprimes.
- A320656 into squarefree semiprimes.
- A320732 into primes or semiprimes.
- A322353 into distinct semiprimes.
- A339661 into distinct squarefree semiprimes.
- A339742 into distinct primes or squarefree semiprimes.
- A339839 into distinct primes or semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A321728 is conjectured to count non-half-loop-graphical partitions of n.
- A339617 counts non-graphical partitions of 2n, ranked by A339618.
- A339655 counts non-loop-graphical partitions of 2n (A339657).
Cf. A000070, A028260, A320893, A320922, A339741, A339846.

filter:= proc(n)
  g(map(t -> t[2], ifactors(n)[2]))
end proc;
g:= proc(L) option remember; local x,i,j,t,s,Cons,R;
  if nops(L) = 1 then return L[1] > 3
  elif nops(L) = 2 then return max(L) > 4
  fi;
  Cons:= {seq(x[i] + x[i,i] + add(x[j,i], j=1..i-1)
     + add(x[i,j],j=i+1..nops(L)) = L[i], i=1..nops(L))};
  R:= traperror(Optimization:-LPSolve(0,Cons, assume=binary));
  type(R,string)
end proc:
select(filter, [$2..2000]); # Robert Israel, Dec 28 2020

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

A339889 Products of distinct primes or semiprimes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70

Offset: 1

Gus Wiseman, Dec 23 2020

nonn

Numbers that can be factored into distinct primes or semiprimes.

A semiprime (A001358) is a product of any two prime numbers.

			See A339840 for examples.

Robert Israel, Table of n, a(n) for n = 1..10000
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

See link for additional cross-references.
Allowing only primes gives A005117.
Not allowing squares of primes gives A339741.
Positions of nonzeros in A339839.
Complement of A339840.
A001055 counts factorizations.
A001358 lists semiprimes, with squarefree case A006881.
A320663 counts non-isomorphic multiset partitions into singletons or pairs.
A320732 counts factorizations into primes or semiprimes.
A339742 counts factorizations into distinct primes or squarefree semiprimes.
A339841 have exactly one factorization into primes or semiprimes.
Cf. A002100, A013929, A293511, A320893, A339740, A339846.

N:= 100: # for terms <= N
B:= select(t -> numtheory:-bigomega(t) <= 2, {$2..N}):
S:= {1}:
for b in B do
  S:= S union map(`*`,select(`<=`,S,N/b),b)
od:
sort(convert(S,list)); # Robert Israel, Dec 28 2020

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Select[facs[#],UnsameQ@@#&&SubsetQ[{1,2},PrimeOmega/@#]&]!={}&]

A368726 Number of non-isomorphic connected multiset partitions of weight n into singletons or pairs.

Original entry on oeis.org

1, 1, 3, 3, 8, 10, 26, 38, 93, 161, 381, 732, 1721, 3566, 8369, 18316, 43280, 98401, 234959, 549628, 1327726, 3175670, 7763500, 18905703, 46762513, 115613599, 289185492, 724438500, 1831398264, 4641907993, 11853385002, 30365353560

Offset: 0

Gus Wiseman, Jan 06 2024

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 10 multiset partitions:
  {{1}}  {{1,1}}    {{1},{1,1}}    {{1,1},{1,1}}      {{1},{1,1},{1,1}}
         {{1,2}}    {{2},{1,2}}    {{1,2},{1,2}}      {{1},{1,2},{2,2}}
         {{1},{1}}  {{1},{1},{1}}  {{1,2},{2,2}}      {{2},{1,2},{1,2}}
                                   {{1,3},{2,3}}      {{2},{1,2},{2,2}}
                                   {{1},{1},{1,1}}    {{2},{1,3},{2,3}}
                                   {{1},{2},{1,2}}    {{3},{1,3},{2,3}}
                                   {{2},{2},{1,2}}    {{1},{1},{1},{1,1}}
                                   {{1},{1},{1},{1}}  {{1},{2},{2},{1,2}}
                                                      {{2},{2},{2},{1,2}}
                                                      {{1},{1},{1},{1},{1}}

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

For edges of any size we have A007718.
This is the connected case of A320663.
The case of singletons and strict pairs is A368727, Euler transform A339888.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A007716 counts non-isomorphic multiset partitions, into pairs A007717.
A062740 counts connected loop-graphs, unlabeled A054921.
A320732 counts factorizations into primes or semiprimes, strict A339839.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A001515, A000666, A122848, A283877, A302545, A320462, A321405, A368186, A368598, A368599, A368731.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort/@(#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n], Max@@Length/@#<=2&&Length[csm[#]]<=1&]]],{n,0,8}]

Inverse Euler transform of A320663.

A368727 Number of non-isomorphic connected multiset partitions of weight n into singletons or strict pairs.

Original entry on oeis.org

1, 1, 2, 2, 5, 6, 15, 21, 49, 82, 184, 341, 766, 1530, 3428, 7249, 16394, 36009, 82492, 186485, 433096, 1001495, 2358182, 5554644, 13255532, 31718030, 76656602, 185982207, 454889643, 1117496012, 2764222322, 6868902152, 17172601190

Offset: 0

Gus Wiseman, Jan 06 2024

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 15 multiset partitions:
  {1}  {12}    {2}{12}    {12}{12}      {2}{12}{12}      {12}{12}{12}
       {1}{1}  {1}{1}{1}  {13}{23}      {2}{13}{23}      {12}{13}{23}
                          {1}{2}{12}    {3}{13}{23}      {13}{23}{23}
                          {2}{2}{12}    {1}{2}{2}{12}    {13}{24}{34}
                          {1}{1}{1}{1}  {2}{2}{2}{12}    {14}{24}{34}
                                        {1}{1}{1}{1}{1}  {1}{2}{12}{12}
                                                         {1}{2}{13}{23}
                                                         {2}{2}{12}{12}
                                                         {2}{2}{13}{23}
                                                         {2}{3}{13}{23}
                                                         {3}{3}{13}{23}
                                                         {1}{1}{2}{2}{12}
                                                         {1}{2}{2}{2}{12}
                                                         {2}{2}{2}{2}{12}
                                                         {1}{1}{1}{1}{1}{1}

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

For edges of any size we have A056156, with loops A007718.
This is the connected case of A339888.
Allowing loops {x,x} gives A368726, Euler transform A320663.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A007716 counts non-isomorphic multiset partitions, into pairs A007717.
A062740 counts connected loop-graphs, unlabeled A054921.
A320732 counts factorizations into primes or semiprimes, strict A339839.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A001515, A000666, A122848, A283877, A302545, A320462, A321405, A368598, A368599, A368731.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]], {s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List /@ c[[1]]],Union@@s[[c[[1]]]]]]]]];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n],And@@UnsameQ@@@#&&Max@@Length/@#<=2&&Length[csm[#]]<=1&]]],{n,0,8}]

Inverse Euler transform of A339888.

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A339742 Number of factorizations of n into distinct primes or squarefree semiprimes.

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A339740 Non-products of distinct primes or squarefree semiprimes.

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A368726 Number of non-isomorphic connected multiset partitions of weight n into singletons or pairs.

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