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(4) = 2 through a(36) = 5 factorizations (showing only the cases where n is a perfect power).
(4) (8) (9) (16) (25) (27) (32) (36)
(2*2) (2*2*2) (3*3) (2*8) (5*5) (3*3*3) (2*2*2*2*2) (4*9)
(4*4) (6*6)
(2*2*2*2) (2*18)
(3*12)
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],IntegerQ[GeometricMean[#]]&]],{n,2,100}]
A326028(n, m=n, facmul=1, facnum=0) = if(1==n,facnum>0 && ispower(facmul,facnum), my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A326028(n/d, d, facmul*d, facnum+1))); (s)); \\ Antti Karttunen, Nov 10 2024
The a(1296) = 4 achiral tree-factorizations are 1296, (36*36), (6*6*6*6), ((6*6)*(6*6)).
a[n_]:=1+Sum[a[d],{d,n^(1/Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]])}];
Array[a,100]
a(n)={my(z, e=ispower(n,,&z)); 1 + if(e, sumdiv(e, d, if(dAndrew Howroyd, Nov 18 2018
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
smakQ[n_]:=And[SameQ@@DeleteCases[primeMS[n],1],And@@smakQ/@DeleteCases[primeMS[n],1]];Select[Range[100],smakQ[#]&]
is(n) = while((n>>=valuation(n,2)) > 1, isprimepower(n,&n) || return(0); n=primepi(n)); 1; \\ Kevin Ryde, Apr 04 2021
The a(12) = 7 ways are: 2^12, 4^6, 8^4, 8^(2^2), 16^3, 64^2, 4096.
a[n_]:=1+Sum[a[g],{g,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}];
Table[a[2^n],{n,100}]
A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409 A294336(n) = if(1==n,n,sumdiv(A052409(n),d,A294336(d))); A294337(n) = sumdiv(n,d,A294336(d)); \\ Or alternatively, after Mathematica-code as: A294337(n) = A294336(2^n); \\ Antti Karttunen, Jun 12 2018
The a(256) = 10 ways are: (2^1)^8 (2^2)^4 (2^4)^2 (2^8)^1 (4^1)^4 (4^2)^2 (4^4)^1 (16^1)^2 (16^2)^1 (256^1)^1
f:= proc(n) local m,d,t; m:= igcd(seq(t[2],t=ifactors(n)[2])); add(numtheory:-tau(d),d=numtheory:-divisors(m)) end proc: f(1):= 1: map(f, [$1..100]); # Robert Israel, Dec 19 2017
Table[Sum[DivisorSigma[0,d],{d,Divisors[GCD@@FactorInteger[n][[All,2]]]}],{n,100}]
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(4) = 9 trees: (1111), ((11)(11)), (((1)(1))((1)(1))), ((1)(1)(1)(1)), (1112), (1122), ((12)(12)), (1123), (1234). The a(6) = 19 trees: (111111), ((111)(111)), (((1)(1)(1))((1)(1)(1))), ((11)(11)(11)), (((1)(1))((1)(1))((1)(1))), ((1)(1)(1)(1)(1)(1)), (111112), (111122), ((112)(112)), (111123), (111222), ((12)(12)(12)), (111223), (111234), (112233), ((123)(123)), (112234), (112345), (123456).
b[n_]:=1+Sum[b[n/d],{d,Rest[Divisors[n]]}];
a[n_]:=Sum[b[GCD@@Length/@Split[ptn]],{ptn,IntegerPartitions[n]}];
Array[a,30]
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}]
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