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Number of partitions of n into 3 parts such that the largest part is relatively prime to at least one other part.

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`%I #6 May 08 2022 16:38:25
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`%S 0,0,1,1,2,2,4,4,6,6,9,8,13,12,16,16,22,18,28,24,31,30,38,32,46,41,52,
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`%T 47,63,48,72,63,78,70,89,72,101,88,107,95,125,96,139,118,141,131,164,
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`%U 128,180,148,184,166,209,161,222,189,228,205,258,192,277,236,279,250,310
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`%N Number of partitions of n into 3 parts such that the largest part is relatively prime to at least one other part.
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`%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
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`%F a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor(1/gcd(n-i-k,k)) + floor(1/gcd(n-i-k,i))).
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`%t Table[Sum[Sum[Sign[Floor[1/GCD[n - i - k, k]] + Floor[1/GCD[n - i - k, i]]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 80}]
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`%t lppQ[{a_,b_,c_}]:=Total[Boole[{CoprimeQ[c,a],CoprimeQ[b,a]}]]>0; Table[Length[ Select[ IntegerPartitions[n,{3}],lppQ]],{n,70}] (* _Harvey P. Dale_, May 08 2022 *)
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`%Y Cf. A340279, A340280.
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`%K nonn
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`%O 1,5
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`%A _Wesley Ivan Hurt_, Jan 02 2021
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