A186971 -id:A186971 - cp's OEIS frontend

A186972 Irregular triangle T(n,k), n>=1, 1<=k<=A186971(n), read by rows: T(n,k) is the number of k-element subsets of {1, 2, ..., n} containing n and having pairwise coprime elements.

1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 4, 5, 2, 1, 2, 1, 1, 6, 11, 8, 2, 1, 4, 6, 4, 1, 1, 6, 12, 10, 3, 1, 4, 5, 2, 1, 10, 31, 42, 26, 6, 1, 4, 6, 4, 1, 1, 12, 45, 79, 72, 33, 6, 1, 6, 14, 16, 9, 2, 1, 8, 21, 25, 14, 3, 1, 8, 24, 36, 29, 12, 2, 1, 16, 79, 183, 228, 157, 56, 8, 1, 6, 15, 20, 15, 6, 1

Offset: 1

Alois P. Heinz, Mar 01 2011

T(n,k) = 0 for k>A186971(n). The triangle contains all positive values of T.

			T(5,3) = 5 because there are 5 3-element subsets of {1,2,3,4,5} containing 5 and having pairwise coprime elements: {1,2,5}, {1,3,5}, {1,4,5}, {2,3,5}, {3,4,5}.
Irregular Triangle T(n,k) begins:
  1;
  1, 1;
  1, 2,  1;
  1, 2,  1;
  1, 4,  5, 2;
  1, 2,  1;
  1, 6, 11, 8, 2;

Alois P. Heinz, Rows n = 1..220, flattened

Columns k=1-10 give: A000012, A000010 (for n>1), A185953, A185348, A186976, A186977, A186978, A186979, A186980, A186981.
Rightmost elements of rows give A186994.
Row sums are A186973.
Cf. A186971.

with(numtheory):
s:= proc(m,r) option remember; mul(`if`(in then 0
    elif k=1 then 1
    elif k=2 and t=n then `if`(n<2, 0, phi(n))
    else c:= 0;
         d:= 2-irem(t,2);
         for h from 1 to n-1 by d do
           if igcd(t, h)=1 then c:= c +b(s(t*h, h), h, k-1) fi
         od; c
      fi
end:
T:= proc(n,k) option remember; b(s(n,n),n,k) end:
seq(seq(T(n, k), k=1..a(n)), n=1..20);

s[m_, r_] := s[m, r] = Product[If[i < r, i, 1], {i, FactorInteger[m][[All, 1]]}]; a[n_] := a[n] = If[n < 4, n, PrimePi[n] - Length[FactorInteger[n]]+2]; b[t_, n_, k_] := b[t, n, k] = Module[{c, d, h}, Which[k == 0 || k > n, 0, k == 1, 1, k == 2 && t == n, If[n < 2, 0, EulerPhi[n]], True, c = 0; d = 2-Mod[t, 2]; For[h = 1, h <= n-1, h = h+d, If[GCD[t, h] == 1, c = c+b[s[t*h, h], h, k-1]]]; c]]; t[n_, k_] := t[n, k] = b[s[n, n], n, k]; Table[Table[t[n, k], {k, 1, a[n]}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)

A186975 Irregular triangle T(n,k), n>=1, 1<=k<=A186971(n), read by rows: T(n,k) is the number of subsets of {1, 2, ..., n} containing n and having <=k pairwise coprime elements.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 3, 4, 1, 5, 10, 12, 1, 3, 4, 1, 7, 18, 26, 28, 1, 5, 11, 15, 16, 1, 7, 19, 29, 32, 1, 5, 10, 12, 1, 11, 42, 84, 110, 116, 1, 5, 11, 15, 16, 1, 13, 58, 137, 209, 242, 248, 1, 7, 21, 37, 46, 48, 1, 9, 30, 55, 69, 72, 1, 9, 33, 69, 98, 110, 112

Offset: 1

Alois P. Heinz, Mar 02 2011

T(n,k) = T(n,k-1) for k>A186971(n). The triangle contains all values of T up to the last element of each row that is different from its predecessor.

			T(5,3) = 10 because there are 10 subsets of {1,2,3,4,5} containing n and having <=3 pairwise coprime elements: {5}, {1,5}, {2,5}, {3,5}, {4,5}, {1,2,5}, {1,3,5}, {1,4,5}, {2,3,5}, {3,4,5}.
Triangle T(n,k) begins:
  1;
  1, 2;
  1, 3, 4;
  1, 3, 4;
  1, 5, 10, 12;
  1, 3, 4;
  1, 7, 18, 26, 28;

Alois P. Heinz, Rows n = 1..200, flattened

Columns k=1-9 give: A000012, A039649 for n>1, A186987, A186988, A186989, A186990, A186991, A186992, A186993.
Rightmost elements of rows give A186973.
Cf. A186971, A186972.

with(numtheory):
s:= proc(m,r) option remember; mul(`if`(in then 0
    elif k=1 then 1
    elif k=2 and t=n then `if`(n<2, 0, phi(n))
    else c:= 0;
         d:= 2-irem(t, 2);
         for h from 1 to n-1 by d do
           if igcd(t, h)=1 then c:= c +b(s(t*h, h), h, k-1) fi
         od; c
      fi
    end:
T:= proc(n, k) option remember;
       b(s(n, n), n, k) +`if`(k=0, 0, T(n, k-1))
    end:
seq(seq(T(n, k), k=1..a(n)), n=1..20);

s[m_, r_] := s[m, r] = Product[If[i < r, i, 1], {i, FactorInteger[m][[All, 1]]}]; a[n_] := a[n] = If[n < 4, n, PrimePi[n]-Length[FactorInteger[n]]+2]; b[t_, n_, k_] := b[t, n, k] = Module[{c, d, h}, Which[k == 0 || k > n, 0, k == 1, 1, k == 2 && t == n, If[n < 2, 0, EulerPhi[n]], True, c = 0; d = 2-Mod[t, 2]; For[h = 1, h <= n-1, h = h+d, If[GCD[t, h] == 1, c = c+b[s[t*h, h], h, k-1] ] ]; c ] ]; t[n_, k_] := t[n, k] = b[s[n, n], n, k]+If[k == 0, 0, t[n, k-1]]; Table[Table[t[n, k], {k, 1, a[n]}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Dec 19 2013, translated from Maple *)

T(n,k) = Sum_{i=1..k} A186972(n,i).

A276187 Number of subsets of {1,..,n} of cardinality >= 2 such that the elements of each counted subset are pairwise coprime.

Original entry on oeis.org

0, 1, 4, 7, 18, 21, 48, 63, 94, 105, 220, 235, 482, 529, 600, 711, 1438, 1501, 3020, 3211, 3594, 3849, 7720, 7975, 11142, 11877, 14628, 15459, 30946, 31201, 62432, 69855, 76126, 80221, 89820, 91611, 183258, 192601, 208600, 214231, 428502, 431573, 863188, 900563

Offset: 1

Robert C. Lyons, Aug 23 2016

nonn

n is prime if and only if a(n) = 2*a(n-1)+n-1. - Robert Israel, Aug 24 2016

			From _Gus Wiseman_, May 08 2021: (Start)
The a(2) = 1 through a(6) = 21 sets:
  {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)

Robert Israel, Table of n, a(n) for n = 1..340

Cf. A000010, A002088, A018805.
The case of pairs is A015614.
The indivisible instead of coprime version is A051026(n) - n.
Allowing empty sets and singletons gives A084422.
The relatively prime instead of pairwise coprime version is A085945(n) - 1.
Allowing all singletons gives A187106.
Allowing only the singleton {1} gives A320426.
Row sums of A320436, each minus one.
The maximal case is counted by A343659.
The version for sets of divisors is A343655(n) - 1.
A000005 counts divisors.
A186972 counts pairwise coprime k-sets containing n.
A186974 counts pairwise coprime k-sets.
A326675 ranks pairwise coprime non-singleton sets.
Cf. A007360, A063647, A186971, A302696, A305713, A320423, A320430, A326077, A327516, A337485.

f:= proc(S) option remember;
    local s, Sp;
    if S = {} then return 1 fi;
    s:= S[-1];
    Sp:= S[1..-2];
    procname(Sp) + procname(select(t -> igcd(t,s)=1, Sp))
end proc:
seq(f({$1..n}) - n - 1, n=1..50); # Robert Israel, Aug 24 2016

f[S_] := f[S] = Module[{s, Sp}, If[S == {}, Return[1]]; s = S[[-1]]; Sp = S[[1;;-2]]; f[Sp] + f[Select[Sp, GCD[#, s] == 1&]]];
Table[f[Range[n]] - n - 1, {n, 1, 50}] (* Jean-François Alcover, Sep 15 2022, after Robert Israel *)

f(n,k=1)=if(n==1, return(2)); if(gcd(k,n)==1, f(n-1,n*k)) + f(n-1,k)
a(n)=f(n)-n-1 \\ Charles R Greathouse IV, Aug 24 2016

from sage.combinat.subsets_pairwise import PairwiseCompatibleSubsets
def is_coprime(x, y): return gcd(x, y) == 1
max_n = 40
seq = []
for n in range(1, max_n+1):
    P = PairwiseCompatibleSubsets(range(1,n+1), is_coprime)
    a_n = len([1 for s in P.list() if len(s) > 1])
    seq.append(a_n)
print(seq)

a(n) = A320426(n) - 1. - Gus Wiseman, May 08 2021

Name and example edited by Robert Israel, Aug 24 2016

A186994 Number of maximal subsets of {1, 2, ..., n} containing n and having pairwise coprime elements.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 6, 1, 6, 2, 3, 2, 8, 1, 8, 2, 4, 2, 8, 1, 8, 4, 8, 6, 24, 1, 24, 6, 10, 6, 15, 2, 30, 6, 10, 3, 30, 2, 30, 6, 5, 6, 30, 2, 30, 6, 20, 12, 60, 4, 30, 6, 20, 12, 60, 2, 60, 12, 10, 12, 36, 4, 72, 12, 24, 3, 72, 4, 72, 12, 12, 12, 36

Offset: 1

Alois P. Heinz, Mar 01 2011

The elements of a maximal subset are 1, n, and powers of primes that have no common factor with n. The cardinalities of maximal subsets is A186971(n).

			a(5) = 2 because there are 2 maximal subsets of {1,2,3,4,5} containing 5 and having pairwise coprime elements: {1,2,3,5}, {1,3,4,5}.
a(9) = 3, the maximal subsets are {1,2,5,7,9}, {1,4,5,7,9}, {1,5,7,8,9}.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A186971. Rightmost elements in rows of A186972.

with(numtheory):
a:= n-> mul(ilog[j](n), j={ithprime(i)$i=1..pi(n)} minus factorset(n)):
seq(a(n), n=1..200);

a[n_] := Product[Log[p, n] // Floor, {p, Select[Range[n-1], PrimeQ[#] && GCD[n, #] == 1&]}]; Table[a[n], {n, 1, 200}] (* Jean-François Alcover, Dec 09 2014, after Alois P. Heinz *)

a(n) = Product_{p in Primes with p

A186973 Number of subsets of {1, 2, ..., n} containing n and having pairwise coprime elements; also row sums of A186972.

Original entry on oeis.org

1, 2, 4, 4, 12, 4, 28, 16, 32, 12, 116, 16, 248, 48, 72, 112, 728, 64, 1520, 192, 384, 256, 3872, 256, 3168, 736, 2752, 832, 15488, 256, 31232, 7424, 6272, 4096, 9600, 1792, 91648, 9344, 16000, 5632, 214272, 3072, 431616, 37376, 38912, 43008, 982528

Offset: 1

Alois P. Heinz, Mar 01 2011

nonn

			a(6) = 4 because there are 4 subsets of {1,2,3,4,5,6} containing 6 and having pairwise coprime elements: {6}, {1,6}, {5,6}, {1,5,6}.

Alois P. Heinz, Table of n, a(n) for n = 1..220

Cf. A186971, A186972, A186994. Rightmost elements in rows of triangle A186975.

with(numtheory):
s:= proc(m, r) option remember; mul(`if`(i mul(ilog[j](n), j={ithprime(i)$i=1..pi(n)} minus factorset(n)):
b:= proc(t, n, k) option remember; local c, d, h;
      if k=0 or k>n then 0
    elif k=1 then 1
    elif k=2 and t=n then `if`(n<2, 0, phi(n))
    else c:= 0;
         d:= 2-irem(t, 2);
         for h from 1 to n-1 by d do
           if igcd(t, h)=1 then c:= c +b(s(t*h, h), h, k-1) fi
         end; c
      fi
    end:
a:= n-> h(n) + add(b(s(n, n), n, k), k=1..g(n)-1):
seq(a(n), n=1..50);

s[m_, r_] := s[m, r] = Product[If[in, 0, k == 1, 1, k == 2 && t == n, If[n<2, 0, EulerPhi[n]], True, c=0; d=2-Mod[t, 2]; For[h=1, h <= n-1, h=h+d, If[GCD[t, h] == 1, c=c+b[s[t*h, h], h, k-1]]]; c]]; t[n_, k_] := t[n, k] = b[s[n, n], n, k]; Table[Sum[t[n, k], {k, 1, a[n]}], {n, 1, 50}] (* Jean-François Alcover, Dec 04 2014, after Alois P. Heinz *)

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

Showing 1-6 of 6 results.

cp's OEIS Frontend

A186972 Irregular triangle T(n,k), n>=1, 1<=k<=A186971(n), read by rows: T(n,k) is the number of k-element subsets of {1, 2, ..., n} containing n and having pairwise coprime elements.

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A186975 Irregular triangle T(n,k), n>=1, 1<=k<=A186971(n), read by rows: T(n,k) is the number of subsets of {1, 2, ..., n} containing n and having <=k pairwise coprime elements.

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A276187 Number of subsets of {1,..,n} of cardinality >= 2 such that the elements of each counted subset are pairwise coprime.

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A186994 Number of maximal subsets of {1, 2, ..., n} containing n and having pairwise coprime elements.

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A186973 Number of subsets of {1, 2, ..., n} containing n and having pairwise coprime elements; also row sums of A186972.

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Maple

Mathematica

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

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