A322353 Number of factorizations of n into distinct semiprimes; a(1) = 1 by convention.

1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 2, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 2, 1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0

Offset: 1

Author

Antti Karttunen, Dec 06 2018

Keywords

• nonn

Comments

A semiprime (A001358) is a product of any two prime numbers. In the even case, these factorizations have A001222(n)/2 factors. - Gus Wiseman, Dec 31 2020

Records 1, 2, 3, 4, 5, 9, 13, 15, 17, ... occur at 1, 60, 210, 840, 1260, 4620, 27720, 30030, 69300, ...

Examples

			a(4) = 1, as there is just one way to factor 4 into distinct semiprimes, namely as {4}.
From _Gus Wiseman_, Dec 31 2020: (Start)
The a(n) factorizations for n = 60, 210, 840, 1260, 4620, 12600, 18480:
  4*15   6*35    4*6*35    4*9*35    4*15*77    4*6*15*35    4*6*10*77
  6*10   10*21   4*10*21   4*15*21   4*21*55    4*6*21*25    4*6*14*55
         14*15   4*14*15   6*10*21   4*33*35    4*9*10*35    4*6*22*35
                 6*10*14   6*14*15   6*10*77    4*9*14*25    4*10*14*33
                           9*10*14   6*14*55    4*10*15*21   4*10*21*22
                                     6*22*35    6*10*14*15   4*14*15*22
                                     10*14*33                6*10*14*22
                                     10*21*22
                                     14*15*22
(End)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000
• Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Unlabeled multiset partitions of this type are counted by A007717.

The version for partitions is A112020, or A101048 without distinctness.

The non-strict version is A320655.

Positions of zeros include A320892.

Positions of nonzero terms are A320912.

The case of squarefree factors is A339661, or A320656 without distinctness.

Allowing prime factors gives A339839, or A320732 without distinctness.

A322661 counts loop-graphs, ranked by A320461.

A001055 counts factorizations, with strict case A045778.

A001358 lists semiprimes, with squarefree case A006881.

A027187 counts partitions of even length, ranked by A028260.

A037143 lists primes and semiprimes.

A338898/A338912/A338913 give the prime indices of semiprimes.

A339846 counts even-length factorizations, with ordered version A174725.

Cf. A001221, A006125, A006129, A028260, A320893, A338915, A339841.

Programs

Mathematica

Table[Count[Subsets[Select[Divisors[n], PrimeOmega[#] == 2 &]], ?(Times @@ # == n &)], {n, 105}] (* _Michael De Vlieger, Dec 11 2020 *)

PARI

A322353(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((2==bigomega(d)&&(d<=m)), s += A322353(n/d, d-1))); (s)); \\ Antti Karttunen, Dec 10 2020

Formula

a(n) = Sum_{d|n} (-1)^A001222(d) * A339839(n/d). - Gus Wiseman, Dec 31 2020