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%I #25 Jun 18 2020 14:03:56
%S 0,0,1,0,0,1,1,1,1,0,1,2,1,0,2,1,2,1,1,1,1,3,1,3,2,2,4,2,3,3,4,1,3,2,
%T 5,4,4,2,6,2,5,2,6,3,6,2,6,4,7,5,7,4,6,5,6,6,8,7,9,5,5,5,11,5,9,3,8,8,
%U 13,8,10,6,6,9,15,13,13,10,15,12,16,14,17,10
%N Number of integer-sided triangles with perimeter n whose side lengths have the same number of divisors.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>
%F a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * [d(k) = d(i) = d(n-i-k)], where [] is the Iverson bracket and d(n) is the number of divisors of n (A000005).
%e a(12) = 2; The perimeter of each of the two triangles [2,5,5] and [4,4,4] is 12 and the side lengths for each triangle share the same number of divisors (i.e., d(2) = d(5) = d(5) = 2 and d(4) = d(4) = d(4) = 3).
%e a(15) = 2; The perimeter of each of the two triangles [3,5,7] and [5,5,5] is 15 and the side lengths for each triangle share the same number of divisors (i.e., d(3) = d(5) = d(7) = 2 and d(5) = d(5) = d(5) = 2).
%t Table[Sum[Sum[KroneckerDelta[DivisorSigma[0, i], DivisorSigma[0, k], DivisorSigma[0, n - i - k]]*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
%Y Cf. A000005, A005044.
%K nonn,easy
%O 1,12
%A _Wesley Ivan Hurt_, Apr 14 2020
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