A100565 -id:A100565 - cp's OEIS frontend

A062319 Number of divisors of n^n, or of A000312(n).

1, 1, 3, 4, 9, 6, 49, 8, 25, 19, 121, 12, 325, 14, 225, 256, 65, 18, 703, 20, 861, 484, 529, 24, 1825, 51, 729, 82, 1653, 30, 29791, 32, 161, 1156, 1225, 1296, 5329, 38, 1521, 1600, 4961, 42, 79507, 44, 4005, 4186, 2209, 48, 9457, 99, 5151, 2704, 5565, 54

Offset: 0

Jason Earls, Jul 05 2001

From Gus Wiseman, May 02 2021: (Start)
Conjecture: The number of divisors of n^n equals the number of pairwise coprime ordered n-tuples of divisors of n. Confirmed up to n = 30. For example, the a(1) = 1 through a(5) = 6 tuples are:
  (1)  (1,1)  (1,1,1)  (1,1,1,1)  (1,1,1,1,1)
       (1,2)  (1,1,3)  (1,1,1,2)  (1,1,1,1,5)
       (2,1)  (1,3,1)  (1,1,1,4)  (1,1,1,5,1)
              (3,1,1)  (1,1,2,1)  (1,1,5,1,1)
                       (1,1,4,1)  (1,5,1,1,1)
                       (1,2,1,1)  (5,1,1,1,1)
                       (1,4,1,1)
                       (2,1,1,1)
                       (4,1,1,1)
The unordered case (pairwise coprime n-multisets of divisors of n) is counted by A343654.
(End)

			From _Gus Wiseman_, May 02 2021: (Start)
The a(1) = 1 through a(5) = 6 divisors:
  1  1  1   1    1
     2  3   2    5
     4  9   4    25
        27  8    125
            16   625
            32   3125
            64
            128
            256
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Harry J. Smith)

Cf. A046798, A077592, A035116. - Enrique Pérez Herrero, Nov 09 2010
Number of divisors of A000312(n).
Taking Omega instead of sigma gives A066959.
Positions of squares are A173339.
Diagonal n = k of the array A343656.
A000005 counts divisors.
A059481 counts k-multisets of elements of {1..n}.
A334997 counts length-k strict chains of divisors of n.
A343658 counts k-multisets of divisors.
Pairwise coprimality:
- A018892 counts coprime pairs of divisors.
- A084422 counts pairwise coprime subsets of {1..n}.
- A100565 counts pairwise coprime triples of divisors.
- A225520 counts pairwise coprime sets of divisors.
- A343652 counts maximal pairwise coprime sets of divisors.
- A343653 counts pairwise coprime non-singleton sets of divisors > 1.
- A343654 counts pairwise coprime sets of divisors > 1.
Cf. A000169, A000272, A002064, A002109, A009998, A048691, A143773, A146291, A176029, A327527, A343657.

[NumberOfDivisors(n^n): n in  [0..60]]; // Vincenzo Librandi, Nov 09 2014

A062319[n_IntegerQ]:=DivisorSigma[0,n^n]; (* Enrique Pérez Herrero, Nov 09 2010 *)
Join[{1},DivisorSigma[0,#^#]&/@Range[60]] (* Harvey P. Dale, Jun 06 2024 *)

je=[]; for(n=0,200,je=concat(je,numdiv(n^n))); je

{ for (n=0, 1000, write("b062319.txt", n, " ", numdiv(n^n)); ) } \\ Harry J. Smith, Aug 04 2009

a(n)=local(fm);fm=factor(n);prod(k=1,matsize(fm)[1],fm[k,2]*n+1) \\ Franklin T. Adams-Watters, May 03 2011

a(n) = if(n==0, 1, sumdiv(n, d, n^omega(d))); \\ Seiichi Manyama, May 12 2021

from math import prod
from sympy import factorint
def A062319(n): return prod(n*d+1 for d in factorint(n).values()) # Chai Wah Wu, Jun 03 2021

a(n) = A000005(A000312(n)). - Enrique Pérez Herrero, Nov 09 2010
a(2^n) = A002064(n). - Gus Wiseman, May 02 2021
a(prime(n)) = prime(n) + 1. - Gus Wiseman, May 02 2021
a(n) = Product_{i=1..s} (1 + n * m_i) where (m_1,...,m_s) is the sequence of prime multiplicities (prime signature) of n. - Gus Wiseman, May 02 2021
a(n) = Sum_{d|n} n^omega(d) for n > 0. - Seiichi Manyama May 12 2021

A320426 Number of nonempty pairwise coprime subsets of {1,...,n}, where a single number is not considered to be pairwise coprime unless it is equal to 1.

Original entry on oeis.org

1, 2, 5, 8, 19, 22, 49, 64, 95, 106, 221, 236, 483, 530, 601, 712, 1439, 1502, 3021, 3212, 3595, 3850, 7721, 7976, 11143, 11878, 14629, 15460, 30947, 31202, 62433, 69856, 76127, 80222, 89821, 91612, 183259, 192602, 208601, 214232, 428503, 431574, 863189

Offset: 1

Gus Wiseman, Jan 08 2019

nonn

Two or more numbers are pairwise coprime if no pair of them has a common divisor > 1.

			The a(4) = 8 subsets of {1,2,3,4} are {1}, {1,2}, {1,3}, {1,4}, {2,3}, {3,4}, {1,2,3}, {1,3,4}. - _Michael B. Porter_, Jan 12 2019
From _Gus Wiseman_, May 09 2021: (Start)
The a(2) = 2 through a(6) = 22 sets:
   {1}     {1}      {1}       {1}        {1}
  {1,2}   {1,2}    {1,2}     {1,2}      {1,2}
          {1,3}    {1,3}     {1,3}      {1,3}
          {2,3}    {1,4}     {1,4}      {1,4}
         {1,2,3}   {2,3}     {1,5}      {1,5}
                   {3,4}     {2,3}      {1,6}
                  {1,2,3}    {2,5}      {2,3}
                  {1,3,4}    {3,4}      {2,5}
                             {3,5}      {3,4}
                             {4,5}      {3,5}
                            {1,2,3}     {4,5}
                            {1,2,5}     {5,6}
                            {1,3,4}    {1,2,3}
                            {1,3,5}    {1,2,5}
                            {1,4,5}    {1,3,4}
                            {2,3,5}    {1,3,5}
                            {3,4,5}    {1,4,5}
                           {1,2,3,5}   {1,5,6}
                           {1,3,4,5}   {2,3,5}
                                       {3,4,5}
                                      {1,2,3,5}
                                      {1,3,4,5}
(End)

The case of pairs is A015614.
The case with singletons is A187106.
The version without singletons (except {1}) is A276187.
Row sums of A320436.
The version for divisors > 1 is A343654.
The version for divisors without singletons is A343655.
The maximal version is A343659.
A018892 counts coprime unordered pairs of divisors.
A051026 counts pairwise indivisible subsets of {1...n}.
A087087 ranks pairwise coprime subsets of {1...n}.
A326675 ranks pairwise coprime non-singleton subsets of {1...n}.
Cf. A007359, A007360, A066620, A089233, A100565, A225520, A325683, A326359, A337485, A343652.

Table[Length[Select[Subsets[Range[n]],CoprimeQ@@#&]],{n,10}]

a(n) = A187106(n) - n + 1 = A084422(n) - n.

a(n) = A276187(n) + 1. - Gus Wiseman, May 08 2021

a(25)-a(43) from Alois P. Heinz, Jan 08 2019

A343652 Number of maximal pairwise coprime sets of divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 8, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 10, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 12, 1, 2, 4, 4, 2, 5, 1, 8, 4, 2, 1, 10, 2, 2

Offset: 1

Gus Wiseman, Apr 25 2021

nonn

Also the number of maximal pairwise coprime sets of divisors > 1 of n. For example, the a(n) sets for n = 12, 30, 36, 60, 120 are:
  {6}    {30}     {6}    {30}     {30}
  {12}   {2,15}   {12}   {60}     {60}
  {2,3}  {3,10}   {18}   {2,15}   {120}
  {3,4}  {5,6}    {36}   {3,10}   {2,15}
         {2,3,5}  {2,3}  {3,20}   {3,10}
                  {2,9}  {4,15}   {3,20}
                  {3,4}  {5,6}    {3,40}
                  {4,9}  {5,12}   {4,15}
                         {2,3,5}  {5,6}
                         {3,4,5}  {5,12}
                                  {5,24}
                                  {8,15}
                                  {2,3,5}
                                  {3,4,5}
                                  {3,5,8}

			The a(n) sets for n = 12, 30, 36, 60, 120:
  {1,6}    {1,30}     {1,6}    {1,30}     {1,30}
  {1,12}   {1,2,15}   {1,12}   {1,60}     {1,60}
  {1,2,3}  {1,3,10}   {1,18}   {1,2,15}   {1,120}
  {1,3,4}  {1,5,6}    {1,36}   {1,3,10}   {1,2,15}
           {1,2,3,5}  {1,2,3}  {1,3,20}   {1,3,10}
                      {1,2,9}  {1,4,15}   {1,3,20}
                      {1,3,4}  {1,5,6}    {1,3,40}
                      {1,4,9}  {1,5,12}   {1,4,15}
                               {1,2,3,5}  {1,5,6}
                               {1,3,4,5}  {1,5,12}
                                          {1,5,24}
                                          {1,8,15}
                                          {1,2,3,5}
                                          {1,3,4,5}
                                          {1,3,5,8}

The case of pairs is A063647.
The case of triples is A066620.
The non-maximal version counting empty sets and singletons is A225520.
The non-maximal version with no 1's is A343653.
The non-maximal version is A343655.
The version for subsets of {1..n} is A343659.
The case without 1's or singletons is A343660.
A018892 counts pairwise coprime unordered pairs of divisors.
A048691 counts pairwise coprime ordered pairs of divisors.
A048785 counts pairwise coprime ordered triples of divisors.
A084422, A187106, A276187, and A320426 count pairwise coprime sets.
A100565 counts pairwise coprime unordered triples of divisors.
A305713 counts pairwise coprime non-singleton strict partitions.
A324837 counts minimal subsets of {1...n} with least common multiple n.
A325683 counts maximal Golomb rulers.
A326077 counts maximal pairwise indivisible sets.
Cf. A005361, A007359, A051026, A062319, A067824, A074206, A146291, A285572, A325859, A326359, A326496, A326675, A343654.

fasmax[y_]:=Complement[y,Union@@Most@*Subsets/@y];
Table[Length[fasmax[Select[Subsets[Divisors[n]],CoprimeQ@@#&]]],{n,100}]

a(n) = A343660(n) + A005361(n).

A343655 Number of pairwise coprime sets of divisors of n, where a singleton is not considered pairwise coprime unless it is {1}.

Original entry on oeis.org

1, 2, 2, 3, 2, 6, 2, 4, 3, 6, 2, 10, 2, 6, 6, 5, 2, 10, 2, 10, 6, 6, 2, 14, 3, 6, 4, 10, 2, 22, 2, 6, 6, 6, 6, 17, 2, 6, 6, 14, 2, 22, 2, 10, 10, 6, 2, 18, 3, 10, 6, 10, 2, 14, 6, 14, 6, 6, 2, 38, 2, 6, 10, 7, 6, 22, 2, 10, 6, 22, 2, 24, 2, 6, 10, 10, 6, 22, 2

Offset: 1

Gus Wiseman, Apr 26 2021

nonn

First differs from A015995 at a(210) = 88, A015995(210) = 86.

			For example, the a(n) subsets for n = 1, 2, 4, 6, 8, 12, 16, 24 are:
  {1}  {1}    {1}    {1}      {1}    {1}      {1}     {1}
       {1,2}  {1,2}  {1,2}    {1,2}  {1,2}    {1,2}   {1,2}
              {1,4}  {1,3}    {1,4}  {1,3}    {1,4}   {1,3}
                     {1,6}    {1,8}  {1,4}    {1,8}   {1,4}
                     {2,3}           {1,6}    {1,16}  {1,6}
                     {1,2,3}         {2,3}            {1,8}
                                     {3,4}            {2,3}
                                     {1,12}           {3,4}
                                     {1,2,3}          {3,8}
                                     {1,3,4}          {1,12}
                                                      {1,24}
                                                      {1,2,3}
                                                      {1,3,4}
                                                      {1,3,8}

The case of pairs is A063647.
The case of triples is A066620.
The version with empty sets and singletons is A225520.
A version for prime indices is A304711.
The version for strict integer partitions is A305713.
The version for subsets of {1..n} is A320426 = A276187 + 1.
The version for binary indices is A326675.
The version for integer partitions is A327516.
The version for standard compositions is A333227.
The maximal case is A343652.
The case without 1's is A343653.
The case without 1's with singletons is A343654.
The maximal case without 1's is A343660.
A018892 counts coprime unordered pairs of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A100565 counts pairwise coprime unordered triples of divisors.
A325683 counts maximal Golomb rulers.
A326077 counts maximal pairwise indivisible sets.
Cf. A007360, A062319, A067824, A076078, A084422, A187106, A282935, A285572, A304709, A320423, A337485, A343659.

Table[Length[Select[Subsets[Divisors[n]],CoprimeQ@@#&]],{n,100}]

A066620 Number of unordered triples of distinct pairwise coprime divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 7, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 7, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 13, 0, 1, 2, 0, 1, 7, 0, 2, 1, 7, 0, 6, 0, 1, 2, 2, 1, 7, 0, 4, 0, 1, 0, 13, 1, 1, 1, 3, 0, 13, 1, 2, 1, 1, 1, 5, 0, 2, 2, 4, 0, 7, 0

Offset: 1

K. B. Subramaniam (kb_subramaniambalu(AT)yahoo.com) and Amarnath Murthy, Dec 24 2001

nonn

a(m) = a(n) if m and n have same factorization structure.

			a(24) = 3: the divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. The triples are (1, 2, 3), (1, 2, 9), (1, 3, 4).
a(30) = 7: the triples are (1, 2, 3), (1, 2, 5), (1, 3, 5), (2, 3, 5), (1, 3, 10), (1, 5, 6), (1, 2, 15).

Amarnath Murthy, Decomposition of the divisors of a natural number into pairwise coprime sets, Smarandache Notions Journal, vol. 12, No. 1-2-3, Spring 2001.pp 303-306.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Positions of zeros are A000961.
Positions of ones are A006881.
The version for subsets of {1..n} instead of divisors is A015617.
The non-strict ordered version is A048785.
The version for pairs of divisors is A063647.
The non-strict version (3-multisets) is A100565.
The version for partitions is A220377 (non-strict: A307719).
A version for sets of divisors of any size is A225520.
A000005 counts divisors.
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A007304 ranks 3-part strict partitions.
A014311 ranks 3-part compositions.
A014612 ranks 3-part partitions.
A018892 counts unordered pairs of coprime divisors (ordered: A048691).
A051026 counts pairwise indivisible subsets of {1..n}.
A337461 counts 3-part pairwise coprime compositions.
A338331 lists Heinz numbers of pairwise coprime partitions.
Cf. A007360, A023022, A084422, A276187, A282935, A305713, A337563, A337605, A343652, A343655.

Table[Length[Select[Subsets[Divisors[n],{3}],CoprimeQ@@#&]],{n,100}] (* Gus Wiseman, Apr 28 2021 *)

A066620(n) = (numdiv(n^3)-3*numdiv(n)+2)/6; \\ After Jovovic's formula. - Antti Karttunen, May 27 2017

from sympy import divisor_count as d
def a(n): return (d(n**3) - 3*d(n) + 2)/6 # Indranil Ghosh, May 27 2017

In the reference it is shown that if k is a squarefree number with r prime factors and m with (r+1) prime factors then a(m) = 4*a(k) + 2^k - 1.

a(n) = (tau(n^3)-3*tau(n)+2)/6. - Vladeta Jovovic, Nov 27 2004

More terms from Vladeta Jovovic, Apr 03 2003
Name corrected by Andrey Zabolotskiy, Dec 09 2020
Name corrected by Gus Wiseman, Apr 28 2021 (ordered version is 6*a(n))

A343654 Number of pairwise coprime sets of divisors > 1 of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 4, 3, 5, 2, 8, 2, 5, 5, 5, 2, 8, 2, 8, 5, 5, 2, 11, 3, 5, 4, 8, 2, 15, 2, 6, 5, 5, 5, 13, 2, 5, 5, 11, 2, 15, 2, 8, 8, 5, 2, 14, 3, 8, 5, 8, 2, 11, 5, 11, 5, 5, 2, 25, 2, 5, 8, 7, 5, 15, 2, 8, 5, 15, 2, 18, 2, 5, 8, 8, 5, 15, 2, 14, 5, 5

Offset: 1

Gus Wiseman, Apr 26 2021

nonn

First differs from A100565 at a(210) = 52, A100565(210) = 51.

			The a(n) sets for n = 1, 2, 4, 6, 8, 12, 24, 30, 32, 36, 48:
  {}  {}   {}   {}     {}   {}     {}     {}       {}    {}     {}
      {2}  {2}  {2}    {2}  {2}    {2}    {2}      {2}   {2}    {2}
           {4}  {3}    {4}  {3}    {3}    {3}      {4}   {3}    {3}
                {6}    {8}  {4}    {4}    {5}      {8}   {4}    {4}
                {2,3}       {6}    {6}    {6}      {16}  {6}    {6}
                            {12}   {8}    {10}     {32}  {9}    {8}
                            {2,3}  {12}   {15}           {12}   {12}
                            {3,4}  {24}   {30}           {18}   {16}
                                   {2,3}  {2,3}          {36}   {24}
                                   {3,4}  {2,5}          {2,3}  {48}
                                   {3,8}  {3,5}          {2,9}  {2,3}
                                          {5,6}          {3,4}  {3,4}
                                          {2,15}         {4,9}  {3,8}
                                          {3,10}                {3,16}
                                          {2,3,5}

The version for partitions is A007359.
The version for subsets of {1..n} is A084422.
The case of pairs is A089233.
The version with 1's is A225520.
The maximal case is A343652.
The case without empty sets or singletons is A343653.
The maximal case without singletons is A343660.
A018892 counts pairwise coprime unordered pairs of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A100565 counts pairwise coprime unordered triples of divisors.
A187106, A276187, and A320426 count other types of pairwise coprime sets.
A326077 counts maximal pairwise indivisible sets.
Cf. A007360, A051026, A062319, A074206, A087087, A101268, A285572, A305713, A320423, A326675, A337485, A343655.

pwcop[y_]:=And@@(GCD@@#1==1&)/@Subsets[y,{2}];
Table[Length[Select[Subsets[Rest[Divisors[n]]],pwcop]],{n,100}]

A343659 Number of maximal pairwise coprime subsets of {1..n}.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 7, 9, 9, 10, 10, 12, 16, 19, 19, 20, 20, 22, 28, 32, 32, 33, 54, 61, 77, 84, 84, 85, 85, 94, 112, 123, 158, 161, 161, 176, 206, 212, 212, 214, 214, 229, 241, 260, 260, 263, 417, 428, 490, 521, 521, 526, 655, 674, 764, 818, 818, 820, 820, 874, 918, 975, 1182, 1189, 1189

Offset: 1

Gus Wiseman, Apr 26 2021

nonn

For this sequence, it does not matter whether singletons are considered pairwise coprime.
For n > 2, also the number of maximal pairwise coprime subsets of {2..n}.
For each prime p <= n, p divides exactly one element of each maximal subset. - Bert Dobbelaere, May 04 2021

			The a(1) = 1 through a(9) = 7 subsets:
  {1}  {12}  {123}  {123}  {1235}  {156}   {1567}   {1567}   {1567}
                    {134}  {1345}  {1235}  {12357}  {12357}  {12357}
                                   {1345}  {13457}  {13457}  {12579}
                                                    {13578}  {13457}
                                                             {13578}
                                                             {14579}
                                                             {15789}

Bert Dobbelaere, Table of n, a(n) for n = 1..500
Bert Dobbelaere, Python program

The case of pairs is A015614.
The case of triples is A015617.
The non-maximal version counting empty sets and singletons is A084422.
The non-maximal version counting singletons is A187106.
The non-maximal version is A320426(n) = A276187(n) + 1.
The version for indivisibility instead of coprimality is A326077.
The version for sets of divisors is A343652.
The version for sets of divisors > 1 is A343660.
A018892 counts coprime unordered pairs of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A100565 counts pairwise coprime unordered triples of divisors.
Cf. A007360, A067824, A087087, A225520, A324837, A325683, A325859, A326358, A326496, A326675, A333227, A343653, A343655.

fasmax[y_]:=Complement[y,Union@@Most@*Subsets/@y];
Table[Length[fasmax[Select[Subsets[Range[n]],CoprimeQ@@#&]]],{n,15}]

More terms from Bert Dobbelaere, May 04 2021

A343653 Number of non-singleton pairwise coprime nonempty sets of divisors > 1 of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 7, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 7, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 13, 0, 1, 2, 0, 1, 7, 0, 2, 1, 7, 0, 6, 0, 1, 2, 2, 1, 7, 0, 4, 0, 1, 0, 13, 1, 1

Offset: 1

Gus Wiseman, Apr 25 2021

nonn

First differs from A066620 at a(210) = 36, A066620(210) = 35.

			The a(n) sets for n = 6, 12, 24, 30, 36, 60, 72, 96:
  {2,3}  {2,3}  {2,3}  {2,3}    {2,3}  {2,3}    {2,3}  {2,3}
         {3,4}  {3,4}  {2,5}    {2,9}  {2,5}    {2,9}  {3,4}
                {3,8}  {3,5}    {3,4}  {3,4}    {3,4}  {3,8}
                       {5,6}    {4,9}  {3,5}    {3,8}  {3,16}
                       {2,15}          {4,5}    {4,9}  {3,32}
                       {3,10}          {5,6}    {8,9}
                       {2,3,5}         {2,15}
                                       {3,10}
                                       {3,20}
                                       {4,15}
                                       {5,12}
                                       {2,3,5}
                                       {3,4,5}

The case of pairs is A089233.
The version with 1's, empty sets, and singletons is A225520.
The version for subsets of {1..n} is A320426.
The version for strict partitions is A337485.
The version for compositions is A337697.
The version for prime indices is A337984.
The maximal case with 1's is A343652.
The version with empty sets is a(n) + 1.
The version with singletons is A343654(n) - 1.
The version with empty sets and singletons is A343654.
The version with 1's is A343655.
The maximal case is A343660.
A018892 counts pairwise coprime unordered pairs of divisors.
A048691 counts pairwise coprime ordered pairs of divisors.
A048785 counts pairwise coprime ordered triples of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A100565 counts pairwise coprime unordered triples of divisors.
A305713 counts pairwise coprime non-singleton strict partitions.
A343659 counts maximal pairwise coprime subsets of {1..n}.
Cf. A007359, A067824, A074206, A076078, A084422, A187106, A285572, A324837, A326675, A327516, A338315.

Table[Length[Select[Subsets[Rest[Divisors[n]]],CoprimeQ@@#&]],{n,100}]

A343660 Number of maximal pairwise coprime sets of at least two divisors > 1 of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 4, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 4, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 8, 0, 1, 2, 0, 1, 4, 0, 2, 1, 4, 0, 6, 0, 1, 2, 2, 1, 4, 0, 4, 0, 1, 0, 8, 1, 1, 1

Offset: 1

Gus Wiseman, Apr 26 2021

nonn

			The a(n) sets for n = 6, 12, 24, 30, 36, 60, 72, 96:
  {2,3}  {2,3}  {2,3}  {5,6}    {2,3}  {5,6}    {2,3}  {2,3}
         {3,4}  {3,4}  {2,15}   {2,9}  {2,15}   {2,9}  {3,4}
                {3,8}  {3,10}   {3,4}  {3,10}   {3,4}  {3,8}
                       {2,3,5}  {4,9}  {3,20}   {3,8}  {3,16}
                                       {4,15}   {4,9}  {3,32}
                                       {5,12}   {8,9}
                                       {2,3,5}
                                       {3,4,5}

The case of pairs is A089233.
The case with 1's is A343652.
The case with singletons is (also) A343652.
The non-maximal version is A343653.
The non-maximal version with 1's is A343655.
The version for subsets of {2..n} is A343659 (for n > 2).
A018892 counts coprime unordered pairs of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A066620 counts pairwise coprime 3-sets of divisors.
A100565 counts pairwise coprime unordered triples of divisors.
Cf. A005361, A007359, A007360, A067824, A074206, A225520, A276187, A320426, A325683, A326077, A326359, A326496, A337485, A343654.

fasmax[y_]:=Complement[y,Union@@Most@*Subsets/@y];
Table[Length[fasmax[Select[Subsets[Rest[Divisors[n]]],CoprimeQ@@#&]]],{n,100}]

a(n) = A343652(n) - A005361(n).

cp's OEIS Frontend

A062319 Number of divisors of n^n, or of A000312(n).

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A320426 Number of nonempty pairwise coprime subsets of {1,...,n}, where a single number is not considered to be pairwise coprime unless it is equal to 1.

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A343652 Number of maximal pairwise coprime sets of divisors of n.

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A343655 Number of pairwise coprime sets of divisors of n, where a singleton is not considered pairwise coprime unless it is {1}.

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A066620 Number of unordered triples of distinct pairwise coprime divisors of n.

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A343654 Number of pairwise coprime sets of divisors > 1 of n.

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A343659 Number of maximal pairwise coprime subsets of {1..n}.

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