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

Gus Wiseman, Dec 23 2020


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.
- A339661 into distinct squarefree semiprimes.
- A339742 into distinct primes or squarefree semiprimes.
- A339839 into distinct primes or semiprimes.
- A339887 into primes or squarefree semiprimes.

The following are sequences of numbers under factorability conditions:
- A320891 cannot be factored into squarefree semiprimes (even omega).
- A320911 can be factored into squarefree semiprimes (even omega).
- A320892 cannot be factored into distinct semiprimes (even omega).
- A320912 can be factored into distinct semiprimes (even omega).
- A320894 cannot be factored into distinct squarefree semiprimes (even omega).
- A339561 can be factored into distinct squarefree semiprimes (even omega).
- A339740 cannot be factored into distinct primes or squarefree semiprimes.
- A339741 can be factored into distinct primes or squarefree semiprimes.
- A339840 cannot be factored into distinct primes or semiprimes.
- A339889 can be factored 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).
- 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).
- A320921 counts connected graphical partitions (A320923).
- A339739 counts non-half-loop-graphical partitions of n.
- A339738 counts half-loop-graphical partitions of n.