A339841 -id:A339841 - cp's OEIS frontend

A339741 Products of distinct primes or squarefree semiprimes.

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84

Offset: 1

Gus Wiseman, Dec 23 2020

nonn

First differs from A212167 in lacking 1080, with prime indices {1,1,1,2,2,2,3}.
First differs from A335433 in lacking 72 (see example).
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 and strict pairs, i.e., into a set of half-loops and edges;
(2) n can be factored into distinct primes or squarefree semiprimes;
(3) the prime signature of n is half-loop-graphical.

			The sequence of terms together with their prime indices begins:
       1: {}           20: {1,1,3}        39: {2,6}
       2: {1}          21: {2,4}          41: {13}
       3: {2}          22: {1,5}          42: {1,2,4}
       5: {3}          23: {9}            43: {14}
       6: {1,2}        26: {1,6}          44: {1,1,5}
       7: {4}          28: {1,1,4}        45: {2,2,3}
      10: {1,3}        29: {10}           46: {1,9}
      11: {5}          30: {1,2,3}        47: {15}
      12: {1,1,2}      31: {11}           50: {1,3,3}
      13: {6}          33: {2,5}          51: {2,7}
      14: {1,4}        34: {1,7}          52: {1,1,6}
      15: {2,3}        35: {3,4}          53: {16}
      17: {7}          36: {1,1,2,2}      55: {3,5}
      18: {1,2,2}      37: {12}           57: {2,8}
      19: {8}          38: {1,8}          58: {1,10}
For example, we have 36 = (2*3*6), so 36 is in the sequence. On the other hand, a complete list of all 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). Since none of these consists of only primes or squarefree semiprimes, 72 is not in the sequence. A complete list of all factorizations of 1080 into primes or squarefree semiprimes is:
  (2*2*2*3*3*3*5)
  (2*2*2*3*3*15)
  (2*2*3*3*3*10)
  (2*2*3*3*5*6)
  (2*2*3*6*15)
  (2*3*3*6*10)
  (2*3*5*6*6)
  (2*6*6*15)
  (3*6*6*10)
  (5*6*6*6)
Since none of these is strict, 1080 is not 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.
Allowing only primes gives A013929.
Not allowing primes gives A339561.
Complement of A339740.
Positions of positive terms in A339742.
Allowing squares of primes gives the complement of A339840.
Unlabeled multiset partitions of this type are counted by A339888.
A001055 counts factorizations.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A339841 have exactly one factorization into primes or semiprimes.
Cf. A001221, A005117, A028260, A030229, A050320, A112798, A309356, A320663, A320893, A320924, A338899.

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[#]!={}&]

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

A339839 Number of factorizations of n into distinct primes or semiprimes.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 0, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 0, 2, 2, 2, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 5, 1, 2, 2, 0, 2, 4, 1, 2, 2, 4, 1, 2, 1, 2, 2, 2, 2, 4, 1, 1, 0, 2, 1, 5, 2, 2, 2, 2, 1, 5, 2, 2, 2, 2, 2, 0, 1, 2, 2, 2, 1, 4, 1, 2, 4

Offset: 1

Gus Wiseman, Dec 20 2020

nonn

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

			The a(n) factorizations for n = 6, 16, 30, 60, 180, 210, 240, 420:
  6    5*6    4*15    4*5*9    6*35     4*6*10    2*6*35
  2*3  2*15   6*10    2*6*15   10*21    2*4*5*6   3*4*35
       3*10   2*5*6   2*9*10   14*15    2*3*4*10  4*5*21
       2*3*5  3*4*5   3*4*15   5*6*7              4*7*15
              2*3*10  3*6*10   2*3*35             5*6*14
                      2*3*5*6  2*5*21             6*7*10
                               2*7*15             2*10*21
                               3*5*14             2*14*15
                               3*7*10             2*5*6*7
                               2*3*5*7            3*10*14
                                                  3*4*5*7
                                                  2*3*5*14
                                                  2*3*7*10

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

A008966 allows only primes.
A320732 is the non-strict version.
A339742 does not allow squares of primes.
A339840 lists the positions of zeros.
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.
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.
- A322353 into distinct semiprimes.
- A339839 [this sequence] into distinct primes or semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A000569 counts graphical partitions (A320922).
- A339656 counts loop-graphical partitions (A339658).
Cf. A000070, A001222, A028260, A320893, A338898, A339617, A339661, A339740, A339741, A339846.

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

A339839(n, u=(1+n)) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (dA339839(n/d, d))); (s)); \\ Antti Karttunen, Feb 10 2023

a(n) = Sum_{d|n squarefree} A322353(n/d).

Data section extended up to a(105) by Antti Karttunen, Feb 10 2023

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[#]=={}&]

A339844 Number of distinct sorted degree sequences among all n-vertex loop-graphs.

Original entry on oeis.org

1, 2, 6, 16, 51, 162, 554, 1918, 6843, 24688, 90342, 333308, 1239725

Offset: 0

Gus Wiseman, Dec 27 2020

In the covering case, these degree sequences, sorted in decreasing order, are the same thing as loop-graphical partitions (A339656). 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.
The following are equivalent characteristics for any positive integer n:
(1) the prime indices of n can be partitioned into distinct pairs, i.e. into a set of loops and edges;
(2) n can be factored into distinct semiprimes;
(3) the prime signature of n is loop-graphical.

			The a(0) = 1 through a(3) = 16 sorted degree sequences:
  ()  (0)  (0,0)  (0,0,0)
      (2)  (0,2)  (0,0,2)
           (1,1)  (0,1,1)
           (1,3)  (0,1,3)
           (2,2)  (0,2,2)
           (3,3)  (0,3,3)
                  (1,1,2)
                  (1,1,4)
                  (1,2,3)
                  (1,3,4)
                  (2,2,2)
                  (2,2,4)
                  (2,3,3)
                  (2,4,4)
                  (3,3,4)
                  (4,4,4)
For example, the loop-graphs
  {{1,1},{2,2},{3,3},{1,2}}
  {{1,1},{2,2},{3,3},{1,3}}
  {{1,1},{2,2},{3,3},{2,3}}
  {{1,1},{2,2},{1,3},{2,3}}
  {{1,1},{3,3},{1,2},{2,3}}
  {{2,2},{3,3},{1,2},{1,3}}
all have degrees y = (3,3,2), 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 without loops is A004251, with covering case A095268.
The half-loop version is A029889, with covering case A339843.
Loop-graphs are counted by A322661 and ranked by A320461 and A340020.
The covering case (no zeros) is A339845.
A007717 counts unlabeled multiset partitions into pairs.
A027187 counts partitions of even length, with Heinz numbers A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A101048 counts partitions into semiprimes.
A339655 counts non-loop-graphical partitions of 2n.
A339656 counts loop-graphical partitions of 2n.
A339659 counts graphical partitions of 2n into k parts.
Cf. A001358, A006125, A006129, A062740, A338898, A339841.

Table[Length[Union[Sort[Table[Count[Join@@#,i],{i,n}]]&/@Subsets[Subsets[Range[n],{1,2}]/.{x_Integer}:>{x,x}]]],{n,0,5}]

a(7)-a(12) from Andrew Howroyd, Jan 10 2024

A339845 Number of distinct sorted degree sequences among all n-vertex loop-graphs without isolated vertices.

Original entry on oeis.org

1, 1, 4, 10, 35, 111, 392, 1364, 4925, 17845, 65654, 242966, 906417

Offset: 0

Gus Wiseman, Dec 27 2020

In the covering case, these degree sequences, sorted in decreasing order, are the same thing as loop-graphical partitions (A339656). 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.
The following are equivalent characteristics for any positive integer n:
(1) the prime indices of n can be partitioned into distinct pairs, i.e. into a set of loops and edges;
(2) n can be factored into distinct semiprimes;
(3) the prime signature of n is loop-graphical.

			The a(0) = 1 through a(3) = 10 sorted degree sequences:
  ()  (2)  (1,1)  (1,1,2)
           (1,3)  (1,1,4)
           (2,2)  (1,2,3)
           (3,3)  (1,3,4)
                  (2,2,2)
                  (2,2,4)
                  (2,3,3)
                  (2,4,4)
                  (3,3,4)
                  (4,4,4)
For example, the loop-graphs
  {{1,1},{2,2},{3,3},{1,2}}
  {{1,1},{2,2},{3,3},{1,3}}
  {{1,1},{2,2},{3,3},{2,3}}
  {{1,1},{2,2},{1,3},{2,3}}
  {{1,1},{3,3},{1,2},{2,3}}
  {{2,2},{3,3},{1,2},{1,3}}
all have degrees y = (3,3,2), 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 without loops is A004251, with covering case A095268.
The half-loop version is A029889, with covering case A339843.
Loop-graphs are counted by A322661 and ranked by A320461 and A340020.
Counting the same partitions by sum gives A339656.
These partitions are ranked by A339658.
The non-covering case (zeros allowed) is A339844.
A007717 counts unlabeled multiset partitions into pairs.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A101048 counts partitions into semiprimes.
A339655 counts non-loop-graphical partitions of 2n.
A339659 counts graphical partitions of 2n into k parts.
Cf. A001358, A006125, A006129, A062740, A338898, A339841.

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

a(n) = A339844(n) - A339844(n-1) for n > 0. - Andrew Howroyd, Jan 10 2024

a(7)-a(12) from Andrew Howroyd, Jan 10 2024

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/@#]&]=={}&]

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

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.

cp's OEIS Frontend

A339741 Products of distinct primes or squarefree semiprimes.

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

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

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A339839 Number of factorizations of n into distinct primes or semiprimes.

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A339740 Non-products of distinct primes or squarefree semiprimes.

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A339844 Number of distinct sorted degree sequences among all n-vertex loop-graphs.

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A339845 Number of distinct sorted degree sequences among all n-vertex loop-graphs without isolated vertices.

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