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A339661 Number of factorizations of n into distinct squarefree semiprimes.

%I A339661 #18 May 02 2022 17:28:23
%S A339661 1,0,0,0,0,1,0,0,0,1,0,0,0,1,1,0,0,0,0,0,1,1,0,0,0,1,0,0,0,0,0,0,1,1,
%T A339661 1,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,1,1,0,1,0,1,0,0,1,0,0,0,
%U A339661 1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,1,1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,1,1,0
%N A339661 Number of factorizations of n into distinct squarefree semiprimes.
%C A339661 A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
%C A339661 Also the number of strict multiset partitions of the multiset of prime factors of n, into distinct strict pairs.
%H A339661 Antti Karttunen, <a href="/A339661/b339661.txt">Table of n, a(n) for n = 1..69300</a>
%H A339661 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A339661 a(n) = Sum_{d|n} (-1)^A001222(d) * A339742(n/d).
%e A339661 The a(n) factorizations for n = 210, 1260, 4620, 30030, 69300 are respectively 3, 2, 6, 15, 7:
%e A339661   (6*35)   (6*10*21)  (6*10*77)   (6*55*91)    (6*10*15*77)
%e A339661   (10*21)  (6*14*15)  (6*14*55)   (6*65*77)    (6*10*21*55)
%e A339661   (14*15)             (6*22*35)   (10*33*91)   (6*10*33*35)
%e A339661                       (10*14*33)  (10*39*77)   (6*14*15*55)
%e A339661                       (10*21*22)  (14*33*65)   (6*15*22*35)
%e A339661                       (14*15*22)  (14*39*55)   (10*14*15*33)
%e A339661                                   (15*22*91)   (10*15*21*22)
%e A339661                                   (15*26*77)
%e A339661                                   (21*22*65)
%e A339661                                   (21*26*55)
%e A339661                                   (22*35*39)
%e A339661                                   (26*33*35)
%e A339661                                   (6*35*143)
%e A339661                                   (10*21*143)
%e A339661                                   (14*15*143)
%t A339661 bfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[bfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
%t A339661 Table[Length[bfacs[n]],{n,100}]
%o A339661 (PARI)
%o A339661 A280710(n) = (bigomega(n)==2*issquarefree(n)); \\ From A280710.
%o A339661 A339661(n, u=(1+n)) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (d<u) && A280710(d), s += A339661(n/d, d))); (s)); \\ _Antti Karttunen_, May 02 2022
%Y A339661 Dirichlet convolution of A008836 (Liouville's lambda) with A339742.
%Y A339661 A050326 allows all squarefree numbers, non-strict case A050320.
%Y A339661 A320656 is the not necessarily strict version.
%Y A339661 A320911 lists all (not just distinct) products of squarefree semiprimes.
%Y A339661 A322794 counts uniform factorizations, such as these.
%Y A339661 A339561 lists positions of nonzero terms.
%Y A339661 A001055 counts factorizations, with strict case A045778.
%Y A339661 A001358 lists semiprimes, with squarefree case A006881.
%Y A339661 A320655 counts factorizations into semiprimes, with strict case A322353.
%Y A339661 The following count vertex-degree partitions and give their Heinz numbers:
%Y A339661 - A000070 counts non-multigraphical partitions of 2n (A339620).
%Y A339661 - A209816 counts multigraphical partitions (A320924).
%Y A339661 - A339655 counts non-loop-graphical partitions of 2n (A339657).
%Y A339661 - A339656 counts loop-graphical partitions (A339658).
%Y A339661 - A339617 counts non-graphical partitions of 2n (A339618).
%Y A339661 - A000569 counts graphical partitions (A320922).
%Y A339661 The following count partitions of even length and give their Heinz numbers:
%Y A339661 - A096373 cannot be partitioned into strict pairs (A320891).
%Y A339661 - A338914 can be partitioned into strict pairs (A320911).
%Y A339661 - A338915 cannot be partitioned into distinct pairs (A320892).
%Y A339661 - A338916 can be partitioned into distinct pairs (A320912).
%Y A339661 - A339559 cannot be partitioned into distinct strict pairs (A320894).
%Y A339661 - A339560 can be partitioned into distinct strict pairs (A339561).
%Y A339661 Cf. A001221, A005117, A007716, A028260, A280710, A300061, A320658, A320659, A320923, A330974.
%K A339661 nonn
%O A339661 1,210
%A A339661 _Gus Wiseman_, Dec 19 2020
%E A339661 More terms and secondary offset added by _Antti Karttunen_, May 02 2022