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A339887
Number of factorizations of n into primes or squarefree semiprimes.
6
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
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
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.
Sequence in context: A318369 A007875 A323437 * A259936 A354870 A050320
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 22 2020
STATUS
approved