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A295920 Number of twice-factorizations of n of type (P,R,R). 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
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.
LINKS
FORMULA
a(n) = Sum_{d|A052409(n)} A000005(A052409(n^(1/d)))^d. - Antti Karttunen, Dec 06 2018, after Mathematica-code
EXAMPLE
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)
MATHEMATICA
Table[Sum[Length[Divisors[GCD@@FactorInteger[n^(1/d)][[All, 2]]]]^d, {d, Divisors[GCD@@FactorInteger[n][[All, 2]]]}], {n, 100}]
PROG
(PARI)
A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
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
CROSSREFS
Sequence in context: A275888 A308166 A295931 * A176187 A180683 A365331
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 30 2017
EXTENSIONS
More terms from Antti Karttunen, Dec 06 2018
STATUS
approved

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Last modified March 28 12:26 EDT 2024. Contains 371254 sequences. (Running on oeis4.)