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%I #17 Jun 22 2022 02:07:52
%S 0,0,0,0,0,1,1,1,1,0,1,1,1,0,2,1,2,0,1,0,1,0,1,1,2,0,3,1,3,0,2,0,2,0,
%T 3,1,3,0,5,1,5,0,4,0,3,0,5,1,5,0,4,0,4,0,2,0,3,0,5,1,3,0,6,1,8,0,5,0,
%U 5,0,4,0,3,0,5,1,6,0,6,0,4,0,7,1,7,0,9,1,10,0
%N Number of integer-sided triangles with perimeter n and prime sides.
%H R. Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>
%F a(n) = A070090(n) + A070092(n) = A070095(n) + A070103(n).
%F a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * c(i) * c(k) * c(n-i-k), where c = A010051. - _Wesley Ivan Hurt_, May 13 2019
%e For n=15 there are A005044(15)=7 integer triangles: [1,7,7], [2,6,7], [3,5,7], [3,6,6], [4,4,7], [4,5,6] and [5,5,5]: two of them consist of primes, therefore a(15)=2.
%t triangleQ[sides_] := With[{s = Total[sides]/2}, AllTrue[sides, # < s&]];
%t a[n_] := Select[IntegerPartitions[n, {3}, Select[Range[Ceiling[n/2]], PrimeQ]], triangleQ] // Length; Array[a, 90] (* _Jean-François Alcover_, Jul 09 2017 *)
%t Table[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[n - i - k] - PrimePi[n - i - k - 1]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] (* _Wesley Ivan Hurt_, May 13 2019 *)
%Y Cf. A070080, A070081, A070082, A070090, A070092, A070095, A070097, A070100, A070103, A070105, A070108, A070111.
%K nonn
%O 1,15
%A _Reinhard Zumkeller_, May 05 2002
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