A327398 Maximum connected squarefree divisor of n.

1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, 2, 17, 3, 19, 5, 21, 11, 23, 3, 5, 13, 3, 7, 29, 5, 31, 2, 11, 17, 7, 3, 37, 19, 39, 5, 41, 21, 43, 11, 5, 23, 47, 3, 7, 5, 17, 13, 53, 3, 11, 7, 57, 29, 59, 5, 61, 31, 21, 2, 65, 11, 67, 17, 23, 7, 71, 3, 73, 37

Offset

1,2

Comments

A squarefree number with prime factorization prime(m_1) * ... * prime(m_k) is connected if the simple labeled graph with vertex set {m_1,...,m_k} and edges between any two vertices with a common divisor greater than 1 is connected. Connected numbers are listed in A305078.

Links

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

Gus Wiseman, Sequences counting and encoding certain classes of multisets

Example

The connected squarefree divisors of 189 are {1, 3, 7, 21}, so a(189) = 21.

Maple

issc:= proc(x) local F, G, E, m, i, j; uses GraphTheory, NumberTheory;
  if not IsSquareFree(x) then return false fi;
  F:= map(pi, ifactors(x)[2][.., 1]); m:= nops(F);
  E:= select(t -> igcd(op(t))>1, {seq(seq([F[i], F[j]], j=i+1..m), i=1..m-1)});
  G:= Graph(F, E);
  IsConnected(G);
end proc:
f:= n -> max(select(issc, NumberTheory:-Divisors(n))):
map(f, [$1..100]); # Robert Israel, Jan 05 2026

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[Less@@#, GCD@@s[[#]]]>1&]}, If[c=={}, s, zsm[Sort[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];
Table[Max[Select[Divisors[n], SquareFreeQ[#]&&Length[zsm[primeMS[#]]]<=1&]], {n, 100}]

Crossrefs

The maximum connected divisor of n is A327076(n).

The maximum squarefree divisor of n is A007947(n).

Connected numbers are A305078.

Connected squarefree numbers are A328513.

Cf. A000005, A001221, A302242, A302494, A304714, A305079, A327656, A328514.

Sequence in context: A324371 A197862 A006530 * A323616 A102095 A331959

Adjacent sequences: A327395 A327396 A327397 * A327399 A327400 A327401

Keyword

nonn,look

Author

Gus Wiseman, Oct 20 2019

Status

approved