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A294068
Number of factorizations of n using perfect powers (elements of A001597) other than 1.
25
1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2
OFFSET
1,16
LINKS
EXAMPLE
The a(1152) = 7 factorizations are (4*4*8*9), (4*8*36), (4*9*32), (8*9*16), (8*144), (9*128), (32*36).
MAPLE
ispp:= proc(n) local F;
F:= ifactors(n)[2];
igcd(op(map(t -> t[2], F)))>1
end proc:
f:= proc(n) local F, np, Q;
F:= map(t -> t[2], ifactors(n)[2]);
np:= mul(ithprime(i)^F[i], i=1..nops(F));
Q:= select(ispp, numtheory:-divisors(np));
G(Q, np)
end proc:
G:= proc(Q, n) option remember; local q, t, k;
if not numtheory:-factorset(n) subset `union`(seq(numtheory:-factorset(q), q=Q)) then return 0 fi;
q:= Q[1]; t:= 0;
for k from 0 while n mod q^k = 0 do
t:= t + procname(Q[2..-1], n/q^k)
od;
t
end proc:
G({}, 1):= 1:
map(f, [$1..200]); # Robert Israel, May 06 2018
MATHEMATICA
ppQ[n_]:=And[n>1, GCD@@FactorInteger[n][[All, 2]]>1];
facsp[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facsp[n/d], Min@@#>=d&]], {d, Select[Divisors[n], ppQ]}]];
Table[Length[facsp[n]], {n, 100}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 05 2018
STATUS
approved