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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

Index entries for sequences computed from exponents in factorization of n

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: A056171 A238949 A076755 * A106490 A122375 A038548

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 June 24 22:01 EDT 2019. Contains 324337 sequences. (Running on oeis4.)