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

1,60

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, ...

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

Formula

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

Example

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)

Mathematica

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

Prog

(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

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.

Sequence in context: A186733 A332042 A171368 * A359785 A133988 A089812

Adjacent sequences: A322350 A322351 A322352 * A322354 A322355 A322356

Keyword

nonn

Author

Antti Karttunen, Dec 06 2018

Status

approved