A294338 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.

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, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Gus Wiseman, Oct 28 2017

Keywords

• nonn

Examples

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

Links

• R. J. Mathar, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Maple

A294338 := proc(n)
    local expo,g,a,d ;
    if n =1 then
        return 1;
    end if;
    # compute gcd of the set of prime power exponents (A052409)
    ifactors(n)[2] ;
    [ seq(op(2,ep),ep=%)] ;
    igcd(op(%)) ;
    # set of divisors of A052409 (without the 1)
    g := numtheory[divisors](%) minus {1} ;
    a := 0 ;
    for d in g do
        # recursive (sort of convolution) call
        a := a+ procname(d)*procname(root[d](n)) ;
    end do:
    1+a ;
end proc:
seq(A294338(n),n=1..120) ; # R. J. Mathar, Nov 27 2017

Mathematica

a[n_]:=1+Sum[a[n^(1/g)]*a[g],{g,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}];
Array[a,100]