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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). 108
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
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
First column of A317748.
Sequence in context: A367098 A343660 A319058 * A335447 A089233 A343653
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 March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)