cp's OEIS Frontend

A336570 Number of maximal sets of proper divisors d|n, d < n, all belonging to A130091 (numbers with distinct prime multiplicities) and forming a divisibility chain.

%I A336570 #16 Sep 02 2020 23:05:22
%S A336570 1,1,1,1,1,2,1,1,1,2,1,2,1,2,2,1,1,2,1,2,2,2,1,3,1,2,1,2,1,3,1,1,2,2,
%T A336570 2,4,1,2,2,3,1,3,1,2,2,2,1,4,1,2,2,2,1,3,2,3,2,2,1,4,1,2,2,1,2,3,1,2,
%U A336570 2,3,1,5,1,2,2,2,2,3,1,4,1,2,1,4,2,2,2
%N A336570 Number of maximal sets of proper divisors d|n, d < n, all belonging to A130091 (numbers with distinct prime multiplicities) and forming a divisibility chain.
%C A336570 A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.
%e A336570 The a(n) sets for n = 36, 120, 144, 180 (ones not shown):
%e A336570   {2,18}    {3,12,24}    {2,18,72}       {2,18}
%e A336570   {3,12}    {5,20,40}    {3,9,18,72}     {3,12}
%e A336570   {2,4,12}  {2,4,8,24}   {3,12,24,48}    {5,20}
%e A336570   {3,9,18}  {2,4,8,40}   {3,12,24,72}    {5,45}
%e A336570             {2,4,12,24}  {2,4,8,16,48}   {2,4,12}
%e A336570             {2,4,20,40}  {2,4,8,24,48}   {2,4,20}
%e A336570                          {2,4,8,24,72}   {3,9,18}
%e A336570                          {2,4,12,24,48}  {3,9,45}
%e A336570                          {2,4,12,24,72}
%t A336570 strsigQ[n_]:=UnsameQ@@Last/@FactorInteger[n];
%t A336570 fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
%t A336570 strses[n_]:=If[n==1,{{}},Join@@Table[Append[#,d]&/@strses[d],{d,Select[Most[Divisors[n]],strsigQ]}]];
%t A336570 Table[Length[fasmax[strses[n]]],{n,100}]
%Y A336570 A336569 is the version for chains containing n.
%Y A336570 A336571 is the non-maximal version.
%Y A336570 A000005 counts divisors.
%Y A336570 A001055 counts factorizations.
%Y A336570 A007425 counts divisors of divisors.
%Y A336570 A032741 counts proper divisors.
%Y A336570 A045778 counts strict factorizations.
%Y A336570 A071625 counts distinct prime multiplicities.
%Y A336570 A074206 counts strict chains of divisors from n to 1.
%Y A336570 A130091 lists numbers with distinct prime multiplicities.
%Y A336570 A181796 counts divisors with distinct prime multiplicities.
%Y A336570 A253249 counts chains of divisors.
%Y A336570 A336422 counts divisible pairs of divisors, both in A130091.
%Y A336570 A336424 counts factorizations using A130091.
%Y A336570 A336500 counts divisors of n in A130091 with quotient also in A130091.
%Y A336570 Cf. A002033, A005117, A098859, A118914, A124010, A305149, A327498, A327523, A336414, A336423, A336425, A336568.
%K A336570 nonn
%O A336570 1,6
%A A336570 _Gus Wiseman_, Jul 29 2020