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

31,11

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

Links

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

Formula

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

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

Example

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)

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 *)

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.

Sequence in context: A182032 A265245 A110270 * A318875 A187143 A187144

Adjacent sequences: A284822 A284823 A284824 * A284826 A284827 A284828

Keyword

nonn,look

Author

Alois P. Heinz, Apr 03 2017

Status

approved