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A281116
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Number of factorizations of n>=2 into factors greater than 1 with no common divisor other than 1 (a(1)=0 by convention).
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108
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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
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OFFSET
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1,12
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COMMENTS
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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
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LINKS
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EXAMPLE
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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).
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MATHEMATICA
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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}]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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