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A281116 Number of factorizations of n>=2 into factors greater than 1 with no common divisor other than 1 (a(1)=0 by convention).

%I A281116 #25 Sep 23 2018 21:25:53
%S A281116 0,0,0,0,0,1,0,0,0,1,0,2,0,1,1,0,0,2,0,2,1,1,0,3,0,1,0,2,0,4,0,0,1,1,
%T A281116 1,5,0,1,1,3,0,4,0,2,2,1,0,5,0,2,1,2,0,3,1,3,1,1,0,8,0,1,2,0,1,4,0,2,
%U A281116 1,4,0,9,0,1,2,2,1,4,0,5,0,1,0,8,1,1,1,3,0,8,1,2,1,1,1,7,0,2,2,5
%N A281116 Number of factorizations of n>=2 into factors greater than 1 with no common divisor other than 1 (a(1)=0 by convention).
%C A281116 Let (e1, e2, ..., ek) be a prime-signature of n (that is, n = p^e1 * q^e2 * ... * r^ek for some primes, p, q, ..., r). Then a(n) is the number of ways of partitioning multiset {e1 x 1, e2 x 2, ..., ek x k} into multisets such that none of the numbers 1 .. k is present in all member multisets of that set partition. - _Antti Karttunen_, Sep 08 2018
%H A281116 Antti Karttunen, <a href="/A281116/b281116.txt">Table of n, a(n) for n = 1..65537</a>
%H A281116 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%e A281116 a(6)=1:  (2*3)
%e A281116 a(12)=2; (2*2*3)       (3*4)
%e A281116 a(24)=3: (2*2*2*3)     (2*3*4)     (3*8)
%e A281116 a(30)=4: (2*3*5)       (2*15)      (3*10)    (5*6)
%e A281116 a(36)=5: (2*2*3*3)     (2*2*9)     (2*3*6)   (3*3*4)   (4*9)
%e A281116 a(96)=7: (2*2*2*2*2*3) (2*2*2*3*4) (2*2*3*8) (2*3*4*4) (2*3*16) (3*4*8) (3*32).
%t A281116 postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A281116 Table[Length[Select[postfacs[n],GCD@@#===1&]],{n,2,100}]
%o A281116 (PARI) A281116(n, m=n, facs=List([])) = if(1==n, (1==gcd(Vec(facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A281116(n/d, d, newfacs))); (s)); \\ _Antti Karttunen_, Sep 08 2018
%Y A281116 Cf. A001055, A007916, A089233, A162247, A259936, A281113, A317751.
%Y A281116 First column of A317748.
%K A281116 nonn
%O A281116 1,12
%A A281116 _Gus Wiseman_, Jan 15 2017
%E A281116 Term a(1) = 0 prepended by _Antti Karttunen_, Sep 08 2018