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

%I A284825 #22 Oct 21 2020 22:49:11
%S A284825 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,
%T A284825 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,
%U A284825 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
%N A284825 Number of partitions of n into 3 parts without common divisors such that every pair of them has common divisors.
%C A284825 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
%H A284825 Alois P. Heinz, <a href="/A284825/b284825.txt">Table of n, a(n) for n = 31..10000</a>
%F A284825 a(n) > 0 iff n in { A230035 }.
%F A284825 a(n) = 0 iff n in { A230034 }.
%e A284825 a(31) = 1: [6,10,15] = [2*3,2*5,3*5].
%e A284825 a(41) = 2: [6,14,21], [6,15,20].
%e A284825 From _Gus Wiseman_, Oct 14 2020: (Start)
%e A284825 Selected terms and the corresponding triples:
%e A284825   a(31)=1: a(41)=2: a(59)=3:  a(77)=4:  a(61)=5:  a(71)=6:
%e A284825 -------------------------------------------------------------
%e A284825   15,10,6  20,15,6  24,20,15  39,26,12  33,22,6   39,26,6
%e A284825            21,14,6  24,21,14  42,20,15  40,15,6   45,20,6
%e A284825                     35,14,10  45,20,12  45,10,6   50,15,6
%e A284825                               50,15,12  28,21,12  35,21,15
%e A284825                                         36,15,10  36,20,15
%e A284825                                                   36,21,14
%e A284825 (End)
%p A284825 a:= proc(n) option remember; add(add(`if`(igcd(i, j)>1
%p A284825       and igcd(i, j, n-i-j)=1 and igcd(i, n-i-j)>1 and
%p A284825       igcd(j, n-i-j)>1, 1, 0), j=i..(n-i)/2), i=2..n/3)
%p A284825     end:
%p A284825 seq(a(n), n=31..137);
%t A284825 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}];
%t A284825 Table[a[n], {n, 31, 137}] (* _Jean-François Alcover_, Jun 13 2018, from Maple *)
%t A284825 stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
%t A284825 Table[Length[Select[IntegerPartitions[n,{3}],GCD@@#==1&&stabQ[#,CoprimeQ]&]],{n,31,100}] (* _Gus Wiseman_, Oct 14 2020 *)
%Y A284825 Cf. A082024, A230034, A230035.
%Y A284825 A023023 does not require pairwise non-coprimality, with strict case A101271.
%Y A284825 A202425 and A328672 count these partitions of any length, ranked by A328868.
%Y A284825 A284825*6 is the ordered version.
%Y A284825 A307719 is the pairwise coprime instead of non-coprime version.
%Y A284825 A337599 does not require relatively primality, with strict case A337605.
%Y A284825 A200976 and A328673 count pairwise non-coprime partitions.
%Y A284825 A289509 gives Heinz numbers of relatively prime partitions.
%Y A284825 A327516 counts pairwise coprime partitions, ranked by A333227.
%Y A284825 A337694 gives Heinz numbers of pairwise non-coprime partitions.
%Y A284825 Cf. A000217, A000741, A001399, A007304, A014612, A220377, A318716, A328679, A337604, A337666.
%K A284825 nonn,look
%O A284825 31,11
%A A284825 _Alois P. Heinz_, Apr 03 2017