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A339661 Number of factorizations of n into distinct squarefree semiprimes. 12
1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1

COMMENTS

A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

Also the number of strict multiset partitions of the multiset of prime factors of n, into distinct strict pairs.

LINKS

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

FORMULA

a(n) = Sum_{d|n} (-1)^A001222(d) * A339742(n/d).

EXAMPLE

The a(n) factorizations for n = 210, 1260, 4620, 30030, 69300:

  (6*35)   (6*10*21)  (6*10*77)   (6*55*91)    (6*10*15*77)

  (10*21)  (6*14*15)  (6*14*55)   (6*65*77)    (6*10*21*55)

  (14*15)             (6*22*35)   (10*33*91)   (6*10*33*35)

                      (10*14*33)  (10*39*77)   (6*14*15*55)

                      (10*21*22)  (14*33*65)   (6*15*22*35)

                      (14*15*22)  (14*39*55)   (10*14*15*33)

                                  (15*22*91)   (10*15*21*22)

                                  (15*26*77)

                                  (21*22*65)

                                  (21*26*55)

                                  (22*35*39)

                                  (26*33*35)

                                  (6*35*143)

                                  (10*21*143)

                                  (14*15*143)

MATHEMATICA

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

Table[Length[bfacs[n]], {n, 100}]

CROSSREFS

A050326 allows all squarefree numbers, non-strict case A050320.

A320656 is the not necessarily strict version.

A320911 lists all (not just distinct) products of squarefree semiprimes.

A322794 counts uniform factorizations, such as these.

A339561 lists positions of nonzero terms.

A001055 counts factorizations, with strict case A045778.

A001358 lists semiprimes, with squarefree case A006881.

A320655 counts factorizations into semiprimes, with strict case A322353.

The following count vertex-degree partitions and give their Heinz numbers:

- A000070 counts non-multigraphical partitions of 2n (A339620).

- A209816 counts multigraphical partitions (A320924).

- 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).

The following count partitions of even length and give their Heinz numbers:

- 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. A001221, A005117, A007716, A028260, A300061, A320658, A320659, A320923, A330974.

Sequence in context: A014184 A014359 A079998 * A320656 A322075 A288220

Adjacent sequences:  A339658 A339659 A339660 * A339662 A339663 A339664

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 19 2020

STATUS

approved

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Last modified July 25 03:53 EDT 2021. Contains 346283 sequences. (Running on oeis4.)