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A339742 Number of factorizations of n into distinct primes or squarefree semiprimes.

%I A339742 #24 May 02 2022 17:28:30
%S A339742 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,
%T A339742 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,
%U A339742 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
%N A339742 Number of factorizations of n into distinct primes or squarefree semiprimes.
%C A339742 A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
%C A339742 The following are equivalent characteristics for any positive integer n:
%C A339742 (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;
%C A339742 (2) n can be factored into distinct primes or squarefree semiprimes.
%H A339742 Antti Karttunen, <a href="/A339742/b339742.txt">Table of n, a(n) for n = 1..69300</a>
%H A339742 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A339742 a(n) = Sum_{d|n squarefree} A339661(n/d).
%e A339742 The a(n) factorizations for n = 6, 30, 60, 210, 420 are respectively 2, 4, 3, 10, 9:
%e A339742   (6)    (5*6)    (6*10)    (6*35)     (2*6*35)
%e A339742   (2*3)  (2*15)   (2*5*6)   (10*21)    (5*6*14)
%e A339742          (3*10)   (2*3*10)  (14*15)    (6*7*10)
%e A339742          (2*3*5)            (5*6*7)    (2*10*21)
%e A339742                             (2*3*35)   (2*14*15)
%e A339742                             (2*5*21)   (2*5*6*7)
%e A339742                             (2*7*15)   (3*10*14)
%e A339742                             (3*5*14)   (2*3*5*14)
%e A339742                             (3*7*10)   (2*3*7*10)
%e A339742                             (2*3*5*7)
%t A339742 sqps[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqps[n/d],Min@@#>d&]],{d,Select[Divisors[n],PrimeQ[#]||SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
%t A339742 Table[Length[sqps[n]],{n,100}]
%o A339742 (PARI)
%o A339742 A353471(n) = (numdiv(n)==2*omega(n));
%o A339742 A339742(n, u=(1+n)) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (d<u) && A353471(d), s += A339742(n/d, d))); (s)); \\ _Antti Karttunen_, May 02 2022
%Y A339742 Dirichlet convolution of A008966 with A339661.
%Y A339742 A008966 allows only primes.
%Y A339742 A339661 does not allow primes, only squarefree semiprimes.
%Y A339742 A339740 lists the positions of zeros.
%Y A339742 A339741 lists the positions of positive terms.
%Y A339742 A339839 allows nonsquarefree semiprimes.
%Y A339742 A339887 is the non-strict version.
%Y A339742 A001358 lists semiprimes, with squarefree case A006881.
%Y A339742 A002100 counts partitions into squarefree semiprimes.
%Y A339742 A013929 cannot be factored into distinct primes.
%Y A339742 A293511 are a product of distinct squarefree numbers in exactly one way.
%Y A339742 A320663 counts non-isomorphic multiset partitions into singletons or pairs.
%Y A339742 A339840 cannot be factored into distinct primes or semiprimes.
%Y A339742 A339841 have exactly one factorization into primes or semiprimes.
%Y A339742 The following count factorizations:
%Y A339742 - A001055 into all positive integers > 1.
%Y A339742 - A050320 into squarefree numbers.
%Y A339742 - A050326 into distinct squarefree numbers.
%Y A339742 - A320655 into semiprimes.
%Y A339742 - A320656 into squarefree semiprimes.
%Y A339742 - A320732 into primes or semiprimes.
%Y A339742 - A322353 into distinct semiprimes.
%Y A339742 - A339742 [this sequence] into distinct primes or squarefree semiprimes.
%Y A339742 - A339839 into distinct primes or semiprimes.
%Y A339742 The following count vertex-degree partitions and give their Heinz numbers:
%Y A339742 - A000569 counts graphical partitions (A320922).
%Y A339742 - A058696 counts all partitions of 2n (A300061).
%Y A339742 - A209816 counts multigraphical partitions (A320924).
%Y A339742 - A339656 counts loop-graphical partitions (A339658).
%Y A339742 -
%Y A339742 The following count partitions/factorizations of even length and give their Heinz numbers:
%Y A339742 - A027187/A339846 has no additional conditions (A028260).
%Y A339742 - A338914/A339562 can be partitioned into edges (A320911).
%Y A339742 - A338916/A339563 can be partitioned into distinct pairs (A320912).
%Y A339742 - A339559/A339564 cannot be partitioned into distinct edges (A320894).
%Y A339742 - A339560/A339619 can be partitioned into distinct edges (A339561).
%Y A339742 Cf. A000070, A001221, A005117, A320893, A320923, A338899, A339113, A339617, A353471.
%K A339742 nonn
%O A339742 1,6
%A A339742 _Gus Wiseman_, Dec 20 2020
%E A339742 More terms from _Antti Karttunen_, May 02 2022