A294338 - OEIS

%I #10 Nov 27 2017 12:07:37

%S 1,1,1,2,1,1,1,2,2,1,1,1,1,1,1,5,1,1,1,1,1,1,1,1,2,1,2,1,1,1,1,2,1,1,

%T 1,2,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,6,1,1,1,1,

%U 1,1,1,1,1,1,1,1,1,1,1,1,5,1,1,1,1,1,1

%N Number of ways to write n as a finite power-tower of positive integers greater than one, allowing both left and right nesting of parentheses.

%H R. J. Mathar, <a href="/A294338/b294338.txt">Table of n, a(n) for n = 1..10000</a>

%e The a(16) = 5 ways are: 16, 4^2, (2^2)^2, 2^4, 2^(2^2).

%p A294338 := proc(n)

%p     local expo,g,a,d ;

%p     if n =1 then

%p         return 1;

%p     end if;

%p     # compute gcd of the set of prime power exponents (A052409)

%p     ifactors(n)[2] ;

%p     [ seq(op(2,ep),ep=%)] ;

%p     igcd(op(%)) ;

%p     # set of divisors of A052409 (without the 1)

%p     g := numtheory[divisors](%) minus {1} ;

%p     a := 0 ;

%p     for d in g do

%p         # recursive (sort of convolution) call

%p         a := a+ procname(d)*procname(root[d](n)) ;

%p     end do:

%p     1+a ;

%p end proc:

%p seq(A294338(n),n=1..120) ; # _R. J. Mathar_, Nov 27 2017

%t a[n_]:=1+Sum[a[n^(1/g)]*a[g],{g,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}];

%t Array[a,100]

%Y Cf. A001597, A007916, A052409, A052410, A089723, A164336, A277562, A284639, A288636, A294336, A294337, A294339.

%K nonn

%O 1,4

%A _Gus Wiseman_, Oct 28 2017