A320426 -id:A320426 - cp's OEIS frontend

A320430 Number of set partitions of [n] where the elements of each non-singleton block are pairwise coprime.

1, 1, 2, 5, 10, 37, 60, 295, 658, 2621, 5368, 38535, 66506, 551529, 1234264, 5004697, 13721836, 143935131, 256835337, 2971237021, 6485081140, 35162930303, 95872321543, 1315397878401, 2399236456202, 25866803180347, 72374386475590, 563368417647305, 1479943119911866

Offset: 0

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) = 10 set partitions: 1|2|3|4, 14|2|3, 13|2|4, 12|3|4, 1|23|4, 1|2|34, 134|2, 123|4, 14|23, 12|34.

Alois P. Heinz, Table of n, a(n) for n = 0..38

Cf. A000110, A000258, A008277, A051424, A085945, A186974, A187106, A302569, A303139, A320424, A320426, A320423, A333517.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[Select[Subsets[Range[n]],Length[#]==1||CoprimeQ@@#&],Range[n]]],{n,10}]

a(14)-a(15) from Alois P. Heinz, Jan 08 2019
a(16) from Alois P. Heinz, Mar 26 2020
a(17)-a(24) from Giovanni Resta, Mar 27 2020
a(25)-a(28) from Alois P. Heinz, Aug 03 2023

A320436 Irregular triangle read by rows where T(n,k) is the number of pairwise coprime k-subsets of {1,...,n}, 1 <= k <= A036234(n), where a single number is not considered to be pairwise coprime unless it is equal to 1.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 2, 1, 9, 7, 2, 1, 11, 8, 2, 1, 17, 19, 10, 2, 1, 21, 25, 14, 3, 1, 27, 37, 24, 6, 1, 31, 42, 26, 6, 1, 41, 73, 68, 32, 6, 1, 45, 79, 72, 33, 6, 1, 57, 124, 151, 105, 39, 6, 1, 63, 138, 167, 114, 41, 6, 1, 71, 159, 192, 128, 44, 6, 1, 79

Offset: 1

Gus Wiseman, Jan 08 2019

			Triangle begins:
   1
   1   1
   1   3   1
   1   5   2
   1   9   7   2
   1  11   8   2
   1  17  19  10   2
   1  21  25  14   3
   1  27  37  24   6
   1  31  42  26   6
   1  41  73  68  32   6
   1  45  79  72  33   6
   1  57 124 151 105  39   6
   1  63 138 167 114  41   6
   1  71 159 192 128  44   6
   1  79 183 228 157  56   8

Except for the k = 1 column, same as A186974.
Row sums are A320426.
Second column is A015614.
Cf. A051424, A084422, A085945, A187106, A276187, A320423, A320430, A320435.

Table[Length[Select[Subsets[Range[n],{k}],CoprimeQ@@#&]],{n,16},{k,PrimePi[n]+1}]

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).

A319187 Number of pairwise coprime subsets of {1,...,n} of maximum cardinality (A036234).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 16, 16, 24, 24, 24, 24, 24, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 72, 72, 72, 72, 72, 72, 72, 72

Offset: 1

Gus Wiseman, Jan 09 2019

nonn

Two or more numbers are pairwise coprime if no pair of them has a common divisor > 1. A single number is not considered to be pairwise coprime unless it is equal to 1.

			The a(8) = 3 subsets are {1,2,3,5,7}, {1,3,4,5,7}, {1,3,5,7,8}.

Ana Rechtman, Décembre 2020, 4e défi (in French), Images des Mathématiques, CNRS, 2020.

Rightmost terms of A186974 and A320436.
Run lengths are A053707.
Cf. A015614, A036234, A051424, A085945, A186971, A186972, A186994, A276187, A303139, A320423, A320426.
Cf. A000005, A045948.

Table[Length[Select[Subsets[Range[n],{PrimePi[n]+1}],CoprimeQ@@#&]],{n,24}] (* see A186974 for a faster program *)

a(n) = prod(p=1, n, if (isprime(p), logint(n, p), 1)); \\ Michel Marcus, Dec 26 2020

a(n) = Product_{p prime <= n} floor(log_p(n)).

a(n) = A000005(A045948(n)). - Ridouane Oudra, Sep 02 2019

cp's OEIS Frontend

A320430 Number of set partitions of [n] where the elements of each non-singleton block are pairwise coprime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A320436 Irregular triangle read by rows where T(n,k) is the number of pairwise coprime k-subsets of {1,...,n}, 1 <= k <= A036234(n), where a single number is not considered to be pairwise coprime unless it is equal to 1.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A319187 Number of pairwise coprime subsets of {1,...,n} of maximum cardinality (A036234).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula