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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). 83
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: A242444 A236441 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 May 26 07:44 EDT 2019. Contains 323579 sequences. (Running on oeis4.)