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A339887 Number of factorizations of n into primes or squarefree semiprimes. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

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

LINKS

Table of n, a(n) for n=1..87.

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

FORMULA

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.

EXAMPLE

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

MATHEMATICA

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}]

CROSSREFS

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.

Sequence in context: A318369 A007875 A323437 * A259936 A050320 A333175

Adjacent sequences:  A339884 A339885 A339886 * A339888 A339889 A339890

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 22 2020

STATUS

approved

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Last modified August 1 00:13 EDT 2021. Contains 346377 sequences. (Running on oeis4.)