A295920 Number of twice-factorizations of n of type (P,R,R).

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

Offset: 1

Author

Gus Wiseman, Nov 30 2017

Keywords

• nonn

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.

Examples

			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)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

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

Programs

Mathematica

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

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

Formula

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

Extensions

More terms from Antti Karttunen, Dec 06 2018