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A295920
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Number of twice-factorizations of n of type (P,R,R).
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5
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1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1
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OFFSET
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1,4
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COMMENTS
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a(n) is also the number of ways to choose a perfect divisor d|n and then a sequence of log_d(n) perfect divisors of d.
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LINKS
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FORMULA
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EXAMPLE
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The a(64) = 17 twice-factorizations are:
(2)*(2)*(2)*(2)*(2)*(2) (2*2)*(2*2)*(2*2) (2*2*2)*(2*2*2) (2*2*2*2*2*2)
(2*2)*(2*2)*(4) (2*2)*(4)*(2*2) (4)*(2*2)*(2*2)
(2*2)*(4)*(4) (4)*(2*2)*(4) (4)*(4)*(2*2)
(2*2*2)*(8) (8)*(2*2*2)
(4)*(4)*(4) (4*4*4)
(8)*(8) (8*8)
(64)
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MATHEMATICA
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Table[Sum[Length[Divisors[GCD@@FactorInteger[n^(1/d)][[All, 2]]]]^d, {d, Divisors[GCD@@FactorInteger[n][[All, 2]]]}], {n, 100}]
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PROG
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(PARI)
A295920(n) = if(1==n, n, my(r); sumdiv(A052409(n), d, if(!ispower(n, d, &r), (1/0), numdiv(A052409(r))^d))); \\ Antti Karttunen, Dec 06 2018, after Mathematica-code
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CROSSREFS
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Cf. A000005, A001055, A052409, A052410, A089723, A279789, A281113, A295923, A295924, A295931, A295935.
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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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