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

1,2

Comments

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.

Links

Table of n, a(n) for n=1..67.

Eric Weisstein's World of Mathematics, Degree Sequence.

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

Example

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.

Mathematica

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[#]!={}&]

Crossrefs

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.

Sequence in context: A335433 A345172 A212167 * A317090 A361393 A179983

Adjacent sequences: A339738 A339739 A339740 * A339742 A339743 A339744

Keyword

nonn

Author

Gus Wiseman, Dec 23 2020

Status

approved