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 A317751 Number of divisors d of n such that there exists a factorization of n into factors > 1 with GCD d. 6
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Also the number of distinct possible GCDs of factorizations of n into factors > 1. Also the number of nonzero terms in row n of A317748. a(prime^n) = A008619(n). If n is squarefree and composite, a(n) = 2. LINKS Antti Karttunen, Table of n, a(n) for n = 1..65537 EXAMPLE The divisors of 36 that are possible GCDs of factorizations of 36 are {1, 2, 3, 6, 36}, so a(36) = 5. MATHEMATICA facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]]; goc[n_, m_]:=Length[Select[facs[n], And[And@@(Divisible[#, m]&/@#), GCD@@(#/m)==1]&]]; Table[Length[Select[Divisors[n], goc[n, #]!=0&]], {n, 100}] PROG (PARI) 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)); A317751(n) = if(1==n, 0, length(A317751aux(n, n, List([]), Set([])))); \\ Antti Karttunen, Sep 08 2018 CROSSREFS Cf. A000005, A000837, A001055, A014963, A045778, A050370, A162247, A281116, A289509. Cf. A317748, A317752, A317755, A317757. Sequence in context: A333749 A238949 A076755 * A106490 A327399 A122375 Adjacent sequences:  A317748 A317749 A317750 * A317752 A317753 A317754 KEYWORD nonn AUTHOR Gus Wiseman, Aug 06 2018 EXTENSIONS More terms from Antti Karttunen, Sep 08 2018 STATUS approved

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Last modified May 7 15:02 EDT 2021. Contains 343650 sequences. (Running on oeis4.)