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

1,4

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 A372288 A180683

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