A339661 - OEIS

%I #18 May 02 2022 17:28:23

%S 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 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 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 Number of factorizations of n into distinct squarefree semiprimes.

%C A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

%C Also the number of strict multiset partitions of the multiset of prime factors of n, into distinct strict pairs.

%H Antti Karttunen, <a href="/A339661/b339661.txt">Table of n, a(n) for n = 1..69300</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(n) = Sum_{d|n} (-1)^A001222(d) * A339742(n/d).

%e The a(n) factorizations for n = 210, 1260, 4620, 30030, 69300 are respectively 3, 2, 6, 15, 7:

%e   (6*35)   (6*10*21)  (6*10*77)   (6*55*91)    (6*10*15*77)

%e   (10*21)  (6*14*15)  (6*14*55)   (6*65*77)    (6*10*21*55)

%e   (14*15)             (6*22*35)   (10*33*91)   (6*10*33*35)

%e                       (10*14*33)  (10*39*77)   (6*14*15*55)

%e                       (10*21*22)  (14*33*65)   (6*15*22*35)

%e                       (14*15*22)  (14*39*55)   (10*14*15*33)

%e                                   (15*22*91)   (10*15*21*22)

%e                                   (15*26*77)

%e                                   (21*22*65)

%e                                   (21*26*55)

%e                                   (22*35*39)

%e                                   (26*33*35)

%e                                   (6*35*143)

%e                                   (10*21*143)

%e                                   (14*15*143)

%t 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 Table[Length[bfacs[n]],{n,100}]

%o (PARI)

%o A280710(n) = (bigomega(n)==2*issquarefree(n)); \\ From A280710.

%o 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 Dirichlet convolution of A008836 (Liouville's lambda) with A339742.

%Y A050326 allows all squarefree numbers, non-strict case A050320.

%Y A320656 is the not necessarily strict version.

%Y A320911 lists all (not just distinct) products of squarefree semiprimes.

%Y A322794 counts uniform factorizations, such as these.

%Y A339561 lists positions of nonzero terms.

%Y A001055 counts factorizations, with strict case A045778.

%Y A001358 lists semiprimes, with squarefree case A006881.

%Y A320655 counts factorizations into semiprimes, with strict case A322353.

%Y The following count vertex-degree partitions and give their Heinz numbers:

%Y - A000070 counts non-multigraphical partitions of 2n (A339620).

%Y - A209816 counts multigraphical partitions (A320924).

%Y - A339655 counts non-loop-graphical partitions of 2n (A339657).

%Y - A339656 counts loop-graphical partitions (A339658).

%Y - A339617 counts non-graphical partitions of 2n (A339618).

%Y - A000569 counts graphical partitions (A320922).

%Y The following count partitions of even length and give their Heinz numbers:

%Y - A096373 cannot be partitioned into strict pairs (A320891).

%Y - A338914 can be partitioned into strict pairs (A320911).

%Y - A338915 cannot be partitioned into distinct pairs (A320892).

%Y - A338916 can be partitioned into distinct pairs (A320912).

%Y - A339559 cannot be partitioned into distinct strict pairs (A320894).

%Y - A339560 can be partitioned into distinct strict pairs (A339561).

%Y Cf. A001221, A005117, A007716, A028260, A280710, A300061, A320658, A320659, A320923, A330974.

%K nonn

%O 1,210

%A _Gus Wiseman_, Dec 19 2020

%E More terms and secondary offset added by _Antti Karttunen_, May 02 2022