A295920 - OEIS

%I #20 Dec 06 2018 16:33:00

%S 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,

%T 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,

%U 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

%N Number of twice-factorizations of n of type (P,R,R).

%C 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.

%H Antti Karttunen, <a href="/A295920/b295920.txt">Table of n, a(n) for n = 1..16384</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(n) = Sum_{d|A052409(n)} A000005(A052409(n^(1/d)))^d. - _Antti Karttunen_, Dec 06 2018, after Mathematica-code

%e The a(64) = 17 twice-factorizations are:

%e (2)*(2)*(2)*(2)*(2)*(2)  (2*2)*(2*2)*(2*2)  (2*2*2)*(2*2*2)  (2*2*2*2*2*2)

%e (2*2)*(2*2)*(4)          (2*2)*(4)*(2*2)    (4)*(2*2)*(2*2)

%e (2*2)*(4)*(4)            (4)*(2*2)*(4)      (4)*(4)*(2*2)

%e (2*2*2)*(8)              (8)*(2*2*2)

%e (4)*(4)*(4)              (4*4*4)

%e (8)*(8)                  (8*8)

%e (64)

%t Table[Sum[Length[Divisors[GCD@@FactorInteger[n^(1/d)][[All,2]]]]^d,{d,Divisors[GCD@@FactorInteger[n][[All,2]]]}],{n,100}]

%o (PARI)

%o A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409

%o 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

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

%K nonn

%O 1,4

%A _Gus Wiseman_, Nov 30 2017

%E More terms from _Antti Karttunen_, Dec 06 2018