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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.
6
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
1,4
LINKS
EXAMPLE
The a(16) = 5 ways are: 16, 4^2, (2^2)^2, 2^4, 2^(2^2).
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]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 28 2017
STATUS
approved