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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

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences computed from exponents in factorization of n

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

Cf. A000005, A001055, A052409, A052410, A089723, A279789, A281113, A295923, A295924, A295931, A295935.

Sequence in context: A275888 A308166 A295931 * A176187 A180683 A214635

Adjacent sequences:  A295917 A295918 A295919 * A295921 A295922 A295923

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 May 28 16:37 EDT 2022. Contains 354119 sequences. (Running on oeis4.)