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.

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, 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, 2, 3, 1, 5, 1, 2, 2, 2, 2, 3, 1, 4, 1, 2, 1, 4, 2, 2, 2

Offset

1,6

Comments

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.

Links

Table of n, a(n) for n=1..87.

Example

The a(n) sets for n = 36, 120, 144, 180 (ones not shown):
  {2,18}    {3,12,24}    {2,18,72}       {2,18}
  {3,12}    {5,20,40}    {3,9,18,72}     {3,12}
  {2,4,12}  {2,4,8,24}   {3,12,24,48}    {5,20}
  {3,9,18}  {2,4,8,40}   {3,12,24,72}    {5,45}
            {2,4,12,24}  {2,4,8,16,48}   {2,4,12}
            {2,4,20,40}  {2,4,8,24,48}   {2,4,20}
                         {2,4,8,24,72}   {3,9,18}
                         {2,4,12,24,48}  {3,9,45}
                         {2,4,12,24,72}

Mathematica

strsigQ[n_]:=UnsameQ@@Last/@FactorInteger[n];
fasmax[y_]:=Complement[y, Union@@(Most[Subsets[#]]&/@y)];
strses[n_]:=If[n==1, {{}}, Join@@Table[Append[#, d]&/@strses[d], {d, Select[Most[Divisors[n]], strsigQ]}]];
Table[Length[fasmax[strses[n]]], {n, 100}]

Crossrefs

A336569 is the version for chains containing n.

A336571 is the non-maximal version.

A000005 counts divisors.

A001055 counts factorizations.

A007425 counts divisors of divisors.

A032741 counts proper divisors.

A045778 counts strict factorizations.

A071625 counts distinct prime multiplicities.

A074206 counts strict chains of divisors from n to 1.

A130091 lists numbers with distinct prime multiplicities.

A181796 counts divisors with distinct prime multiplicities.

A253249 counts chains of divisors.

A336422 counts divisible pairs of divisors, both in A130091.

A336424 counts factorizations using A130091.

A336500 counts divisors of n in A130091 with quotient also in A130091.

Cf. A002033, A005117, A098859, A118914, A124010, A305149, A327498, A327523, A336414, A336423, A336425, A336568.

Sequence in context: A050320 A333175 A294893 * A121382 A305150 A351202

Adjacent sequences: A336567 A336568 A336569 * A336571 A336572 A336573

Keyword

nonn

Author

Gus Wiseman, Jul 29 2020

Status

approved