This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
The a(4096) = 7 ways are: 2^12, 4^6, 8^4, 8^(2^2), 16^3, 64^2, 4096.
The a(16) = 5 ways are: 16, 4^2, (2^2)^2, 2^4, 2^(2^2).
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
a[n_]:=1+Sum[a[n^(1/g)]*a[g],{g,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}];
Array[a,100]
The a(6) = 6 ways are 64, 8^2, (2^3)^2, 4^3, (2^2)^3, 2^6.
f:= proc(n) option remember; local F,t,s,g,a;
F:= ifactors(n)[2];
g:= igcd(op(map(t -> t[2],F)));
t:= 1;
for s in numtheory:-divisors(g) minus {1} do
t:= t + procname(mul(a[1]^(a[2]/s),a=F))*procname(s)
od;
t
end proc:
seq(f(2^n),n=1..100); # Robert Israel, Dec 01 2017
a[n_]:=1+Sum[a[n^(1/g)]*a[g],{g,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}];
Table[a[2^n],{n,100}]
The power tower of 81 is 3^2^2, which turned upside-down is 2^2^3 = 256, so a(81) = 256.
f:= proc(n,r) local F,a,y;
if n = 1 then return 1 fi;
F:= ifactors(n)[2];
y:= igcd(seq(t[2],t=F));
if y = 1 then return n^r fi;
a:= mul(t[1]^(t[2]/y),t=F);
procname(y,a^r)
end proc:
seq(f(n,1),n=1..100); # Robert Israel, May 13 2018
tow[n_]:=If[n==1,{},With[{g=GCD@@FactorInteger[n][[All,2]]},If[g===1,{n},Prepend[tow[g],n^(1/g)]]]];
Table[Power@@Reverse[tow[n]],{n,100}]
We have 16 = 2^2^2, so a(16) = 2. We have 64 = 2^6, so a(64) = 6. We have 81 = 3^2^2, so a(81) = 2. We have 256 = 2^2^3, so a(256) = 3.
a[n_]:=If[n==1,1,With[{g=GCD@@FactorInteger[n][[All,2]]},If[g==1,n,a[g]]]];
Array[a,100]
a(n)={my(t=n); while(t, n=t; t=ispower(t)); n} \\ Andrew Howroyd, Aug 26 2018
We have 64 = 2^6, so a(64) = 2. We have 216 = 6^3, so a(216) = 6. We have 256 = 2^2^3, so a(256) = 2^2 = 4.
tow[n_]:=If[n==1,{},With[{g=GCD@@FactorInteger[n][[All,2]]},If[g===1,{n},Prepend[tow[g],n^(1/g)]]]];
Table[If[n==1,0,Power@@Most[tow[n]]],{n,100}]
A304495(n) = if(1==n,0,my(e, r, tow = List([])); while((e = ispower(n,,&r)) > 1, listput(tow, r); n = e;); n = 1; while(length(tow)>0, e = tow[#tow]; listpop(tow); n = e^n;); (n)); \\ Antti Karttunen, Jul 23 2018
nn=100;
a[n_]:=If[n==1,1,With[{g=GCD@@FactorInteger[n][[All,2]]},If[g==1,n,a[g]]]];
rads=Union[Array[a,nn]];
Table[a[n],{n,nn}]/.Table[rads[[i]]->i,{i,Length[rads]}]
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