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A339839 Number of factorizations of n into distinct primes or semiprimes.

%I A339839 #16 Feb 10 2023 17:11:02
%S A339839 1,1,1,1,1,2,1,1,1,2,1,2,1,2,2,0,1,2,1,2,2,2,1,2,1,2,1,2,1,4,1,0,2,2,
%T A339839 2,2,1,2,2,2,1,4,1,2,2,2,1,1,1,2,2,2,1,2,2,2,2,2,1,5,1,2,2,0,2,4,1,2,
%U A339839 2,4,1,2,1,2,2,2,2,4,1,1,0,2,1,5,2,2,2,2,1,5,2,2,2,2,2,0,1,2,2,2,1,4,1,2,4
%N A339839 Number of factorizations of n into distinct primes or semiprimes.
%C A339839 A semiprime (A001358) is a product of any two prime numbers.
%H A339839 Antti Karttunen, <a href="/A339839/b339839.txt">Table of n, a(n) for n = 1..100000</a>
%H A339839 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A339839 a(n) = Sum_{d|n squarefree} A322353(n/d).
%e A339839 The a(n) factorizations for n = 6, 16, 30, 60, 180, 210, 240, 420:
%e A339839   6    5*6    4*15    4*5*9    6*35     4*6*10    2*6*35
%e A339839   2*3  2*15   6*10    2*6*15   10*21    2*4*5*6   3*4*35
%e A339839        3*10   2*5*6   2*9*10   14*15    2*3*4*10  4*5*21
%e A339839        2*3*5  3*4*5   3*4*15   5*6*7              4*7*15
%e A339839               2*3*10  3*6*10   2*3*35             5*6*14
%e A339839                       2*3*5*6  2*5*21             6*7*10
%e A339839                                2*7*15             2*10*21
%e A339839                                3*5*14             2*14*15
%e A339839                                3*7*10             2*5*6*7
%e A339839                                2*3*5*7            3*10*14
%e A339839                                                   3*4*5*7
%e A339839                                                   2*3*5*14
%e A339839                                                   2*3*7*10
%t A339839 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A339839 Table[Length[Select[facs[n],UnsameQ@@#&&SubsetQ[{1,2},PrimeOmega/@#]&]],{n,100}]
%o A339839 (PARI) A339839(n, u=(1+n)) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (d<u) && (bigomega(d)<3), s += A339839(n/d, d))); (s)); \\ _Antti Karttunen_, Feb 10 2023
%Y A339839 A008966 allows only primes.
%Y A339839 A320732 is the non-strict version.
%Y A339839 A339742 does not allow squares of primes.
%Y A339839 A339840 lists the positions of zeros.
%Y A339839 A001358 lists semiprimes, with squarefree case A006881.
%Y A339839 A002100 counts partitions into squarefree semiprimes.
%Y A339839 A013929 cannot be factored into distinct primes.
%Y A339839 A293511 are a product of distinct squarefree numbers in exactly one way.
%Y A339839 A320663 counts non-isomorphic multiset partitions into singletons or pairs.
%Y A339839 A339841 have exactly one factorization into primes or semiprimes.
%Y A339839 The following count factorizations:
%Y A339839 - A001055 into all positive integers > 1.
%Y A339839 - A320655 into semiprimes.
%Y A339839 - A320656 into squarefree semiprimes.
%Y A339839 - A322353 into distinct semiprimes.
%Y A339839 - A339839 [this sequence] into distinct primes or semiprimes.
%Y A339839 The following count vertex-degree partitions and give their Heinz numbers:
%Y A339839 - A000569 counts graphical partitions (A320922).
%Y A339839 - A339656 counts loop-graphical partitions (A339658).
%Y A339839 Cf. A000070, A001222, A028260, A320893, A338898, A339617, A339661, A339740, A339741, A339846.
%K A339839 nonn
%O A339839 1,6
%A A339839 _Gus Wiseman_, Dec 20 2020
%E A339839 Data section extended up to a(105) by _Antti Karttunen_, Feb 10 2023