A284825 Number of partitions of n into 3 parts without common divisors such that every pair of them has common divisors.

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 3, 0, 5, 0, 0, 0, 1, 0, 5, 0, 1, 0, 6, 0, 6, 0, 0, 0, 4, 0, 6, 0, 0, 0, 9, 0, 2, 1, 2, 0, 9, 0, 8, 1, 1, 0, 5, 0, 14, 0, 1, 0, 15, 0, 14, 0, 0, 1, 14, 0, 14, 0, 2, 0, 15, 0, 6, 1, 2, 1, 11, 0, 18, 1, 1, 0, 10, 0, 23

Offset: 31

Author

Alois P. Heinz, Apr 03 2017

Keywords

• nonn
• look

Comments

The Heinz numbers of these partitions are the intersection of A014612 (triples), A289509 (relatively prime), and A337694 (pairwise non-coprime). - Gus Wiseman, Oct 16 2020

Examples

			a(31) = 1: [6,10,15] = [2*3,2*5,3*5].
a(41) = 2: [6,14,21], [6,15,20].
From _Gus Wiseman_, Oct 14 2020: (Start)
Selected terms and the corresponding triples:
  a(31)=1: a(41)=2: a(59)=3:  a(77)=4:  a(61)=5:  a(71)=6:
-------------------------------------------------------------
  15,10,6  20,15,6  24,20,15  39,26,12  33,22,6   39,26,6
           21,14,6  24,21,14  42,20,15  40,15,6   45,20,6
                    35,14,10  45,20,12  45,10,6   50,15,6
                              50,15,12  28,21,12  35,21,15
                                        36,15,10  36,20,15
                                                  36,21,14
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 31..10000

Crossrefs

Cf. A082024, A230034, A230035.

A023023 does not require pairwise non-coprimality, with strict case A101271.

A202425 and A328672 count these partitions of any length, ranked by A328868.

A284825*6 is the ordered version.

A307719 is the pairwise coprime instead of non-coprime version.

A337599 does not require relatively primality, with strict case A337605.

A200976 and A328673 count pairwise non-coprime partitions.

A289509 gives Heinz numbers of relatively prime partitions.

A327516 counts pairwise coprime partitions, ranked by A333227.

A337694 gives Heinz numbers of pairwise non-coprime partitions.

Cf. A000217, A000741, A001399, A007304, A014612, A220377, A318716, A328679, A337604, A337666.

Programs

Maple

a:= proc(n) option remember; add(add(`if`(igcd(i, j)>1
      and igcd(i, j, n-i-j)=1 and igcd(i, n-i-j)>1 and
      igcd(j, n-i-j)>1, 1, 0), j=i..(n-i)/2), i=2..n/3)
    end:
seq(a(n), n=31..137);

Mathematica

a[n_] := a[n] = Sum[Sum[If[GCD[i, j] > 1 && GCD[i, j, n - i - j] == 1 && GCD[i, n - i - j] > 1 && GCD[j, n - i - j] > 1, 1, 0], {j, i, (n - i)/2} ], {i, 2, n/3}];
Table[a[n], {n, 31, 137}] (* Jean-François Alcover, Jun 13 2018, from Maple *)
stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
Table[Length[Select[IntegerPartitions[n,{3}],GCD@@#==1&&stabQ[#,CoprimeQ]&]],{n,31,100}] (* Gus Wiseman, Oct 14 2020 *)

Formula

a(n) > 0 iff n in { A230035 }.

a(n) = 0 iff n in { A230034 }.