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