A339661 Number of factorizations of n into distinct squarefree semiprimes.

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, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0

Offset: 1

Author

Gus Wiseman, Dec 19 2020

Keywords

• nonn

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.

Examples

			The a(n) factorizations for n = 210, 1260, 4620, 30030, 69300 are respectively 3, 2, 6, 15, 7:
  (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)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..69300
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Dirichlet convolution of A008836 (Liouville's lambda) with A339742.

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, A280710, A300061, A320658, A320659, A320923, A330974.

Programs

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}]

PARI

A280710(n) = (bigomega(n)==2*issquarefree(n)); \\ From A280710.
A339661(n, u=(1+n)) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (dA280710(d), s += A339661(n/d, d))); (s)); \\ Antti Karttunen, May 02 2022

Formula

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

Extensions

More terms and secondary offset added by Antti Karttunen, May 02 2022