A294068 Number of factorizations of n using perfect powers (elements of A001597) other than 1.

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

Robert Israel, Table of n, a(n) for n = 1..10000

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}]

Crossrefs

Positions of zeros are A052485.

Cf. A000961, A001055, A001222, A001597, A001694, A007716, A007916, A045778, A052409, A052410, A052486, A091050, A203025, A303707, A303710.

Sequence in context: A303553 A365550 A188585 * A339553 A181011 A084863

Adjacent sequences: A294065 A294066 A294067 * A294069 A294070 A294071

Keyword

nonn

Author

Gus Wiseman, May 05 2018

Status

approved