A285573 Number of finite nonempty sets of pairwise indivisible divisors of n.

1, 2, 2, 3, 2, 5, 2, 4, 3, 5, 2, 9, 2, 5, 5, 5, 2, 9, 2, 9, 5, 5, 2, 14, 3, 5, 4, 9, 2, 19, 2, 6, 5, 5, 5, 19, 2, 5, 5, 14, 2, 19, 2, 9, 9, 5, 2, 20, 3, 9, 5, 9, 2, 14, 5, 14, 5, 5, 2, 49, 2, 5, 9, 7, 5, 19, 2, 9, 5, 19, 2, 34, 2, 5, 9, 9, 5, 19, 2, 20, 5, 5, 2, 49, 5, 5, 5, 14, 2, 49, 5, 9, 5, 5, 5, 27, 2, 9, 9, 19

Offset: 1

Author

Gus Wiseman, Apr 21 2017

Keywords

• nonn

Comments

From Robert Israel, Apr 21 2017: (Start)

If n = p^k for prime p, a(n) = k+1.

If n = p^j*q^k for distinct primes p,q, a(n) = binomial(j+k+2,j+1)-1. (End)

Examples

			The a(12)=9 sets are: {1}, {2}, {3}, {4}, {6}, {12}, {2,3}, {3,4}, {4,6}.
		

Crossrefs

Cf. A006126, A048143, A076078, A076413, A198085, A285572.

Programs

Maple

g:= proc(S) local x, Sx; option remember;
   if nops(S) = 0 then return {{}} fi;
   x:= S[1];
   Sx:= subsop(1=NULL,S);
   procname(Sx) union map(t -> t union {x}, procname(remove(s -> s mod x = 0 or x mod s = 0, Sx)))
end proc:
f:= proc(n) local F,D;
  F:= ifactors(n)[2];
  D:= numtheory:-divisors(mul(ithprime(i)^F[i,2],i=1..nops(F)));
  nops(g(D)) - 1;
end proc:
map(f, [$1..100]); # Robert Israel, Apr 21 2017

Mathematica

nn=50;
stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Rest[stableSets[Divisors[n],Divisible]]],{n,1,nn}]