cp's OEIS Frontend

A317751 Number of divisors d of n such that there exists a factorization of n into factors > 1 with GCD d.

%I A317751 #14 Sep 12 2018 16:43:57
%S A317751 0,1,1,2,1,2,1,2,2,2,1,3,1,2,2,3,1,3,1,3,2,2,1,3,2,2,2,3,1,2,1,3,2,2,
%T A317751 2,5,1,2,2,3,1,2,1,3,3,2,1,4,2,3,2,3,1,3,2,3,2,2,1,3,1,2,3,4,2,2,1,3,
%U A317751 2,2,1,5,1,2,3,3,2,2,1,4,3,2,1,3,2,2,2,3,1,3,2,3,2,2,2,4,1,3,3,5,1,2,1,3,2
%N A317751 Number of divisors d of n such that there exists a factorization of n into factors > 1 with GCD d.
%C A317751 Also the number of distinct possible GCDs of factorizations of n into factors > 1.
%C A317751 Also the number of nonzero terms in row n of A317748.
%C A317751 a(prime^n) = A008619(n).
%C A317751 If n is squarefree and composite, a(n) = 2.
%H A317751 Antti Karttunen, <a href="/A317751/b317751.txt">Table of n, a(n) for n = 1..65537</a>
%H A317751 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%e A317751 The divisors of 36 that are possible GCDs of factorizations of 36 are {1, 2, 3, 6, 36}, so a(36) = 5.
%t A317751 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A317751 goc[n_,m_]:=Length[Select[facs[n],And[And@@(Divisible[#,m]&/@#),GCD@@(#/m)==1]&]];
%t A317751 Table[Length[Select[Divisors[n],goc[n,#]!=0&]],{n,100}]
%o A317751 (PARI)
%o A317751 A317751aux(n, m, facs, gcds) = if(1==n, setunion([gcd(Vec(facs))],gcds), my(newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); gcds = setunion(gcds,A317751aux(n/d, d, newfacs, gcds)))); (gcds));
%o A317751 A317751(n) = if(1==n,0,length(A317751aux(n, n, List([]), Set([])))); \\ _Antti Karttunen_, Sep 08 2018
%Y A317751 Cf. A000005, A000837, A001055, A014963, A045778, A050370, A162247, A281116, A289509.
%Y A317751 Cf. A317748, A317752, A317755, A317757.
%K A317751 nonn
%O A317751 1,4
%A A317751 _Gus Wiseman_, Aug 06 2018
%E A317751 More terms from _Antti Karttunen_, Sep 08 2018