|
|
A339742
|
|
Number of factorizations of n into distinct primes or squarefree semiprimes.
|
|
12
|
|
|
1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 4, 1, 0, 2, 2, 2, 1, 1, 2, 2, 0, 1, 4, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 3, 1, 2, 1, 0, 2, 4, 1, 1, 2, 4, 1, 0, 1, 2, 1, 1, 2, 4, 1, 0, 0, 2, 1, 3, 2, 2, 2, 0, 1, 3, 2, 1, 2, 2, 2, 0, 1, 1, 1, 1, 1, 4, 1, 0, 4
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,6
|
|
COMMENTS
|
A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct singletons or strict pairs, i.e., into a set of half-loops and edges;
(2) n can be factored into distinct primes or squarefree semiprimes.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = Sum_{d|n squarefree} A339661(n/d).
|
|
EXAMPLE
|
The a(n) factorizations for n = 6, 30, 60, 210, 420 are respectively 2, 4, 3, 10, 9:
(6) (5*6) (6*10) (6*35) (2*6*35)
(2*3) (2*15) (2*5*6) (10*21) (5*6*14)
(3*10) (2*3*10) (14*15) (6*7*10)
(2*3*5) (5*6*7) (2*10*21)
(2*3*35) (2*14*15)
(2*5*21) (2*5*6*7)
(2*7*15) (3*10*14)
(3*5*14) (2*3*5*14)
(3*7*10) (2*3*7*10)
(2*3*5*7)
|
|
MATHEMATICA
|
sqps[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[sqps[n/d], Min@@#>d&]], {d, Select[Divisors[n], PrimeQ[#]||SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
Table[Length[sqps[n]], {n, 100}]
|
|
PROG
|
(PARI)
A353471(n) = (numdiv(n)==2*omega(n));
|
|
CROSSREFS
|
A339661 does not allow primes, only squarefree semiprimes.
A339740 lists the positions of zeros.
A339741 lists the positions of positive terms.
A339839 allows nonsquarefree semiprimes.
A002100 counts partitions into squarefree semiprimes.
A013929 cannot be factored into distinct primes.
A293511 are a product of distinct squarefree numbers in exactly one way.
A320663 counts non-isomorphic multiset partitions into singletons or pairs.
A339840 cannot be factored into distinct primes or semiprimes.
A339841 have exactly one factorization into primes or semiprimes.
The following count factorizations:
- A001055 into all positive integers > 1.
- A050326 into distinct squarefree numbers.
- A320656 into squarefree semiprimes.
- A320732 into primes or semiprimes.
- A322353 into distinct semiprimes.
- A339742 [this sequence] into distinct primes or squarefree semiprimes.
- A339839 into distinct primes or semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
-
The following count partitions/factorizations of even length and give their Heinz numbers:
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|