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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). 105
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,12

COMMENTS

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

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from exponents in factorization of n

EXAMPLE

a(6)=1:  (2*3)

a(12)=2; (2*2*3)       (3*4)

a(24)=3: (2*2*2*3)     (2*3*4)     (3*8)

a(30)=4: (2*3*5)       (2*15)      (3*10)    (5*6)

a(36)=5: (2*2*3*3)     (2*2*9)     (2*3*6)   (3*3*4)   (4*9)

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).

MATHEMATICA

postfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[postfacs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];

Table[Length[Select[postfacs[n], GCD@@#===1&]], {n, 2, 100}]

PROG

(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

CROSSREFS

Cf. A001055, A007916, A089233, A162247, A259936, A281113, A317751.

First column of A317748.

Sequence in context: A236441 A327695 A319058 * A089233 A066620 A219023

Adjacent sequences:  A281113 A281114 A281115 * A281117 A281118 A281119

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 15 2017

EXTENSIONS

Term a(1) = 0 prepended by Antti Karttunen, Sep 08 2018

STATUS

approved

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Last modified October 20 15:07 EDT 2019. Contains 328267 sequences. (Running on oeis4.)