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A070092 Number of isosceles integer triangles with perimeter n and prime side lengths. 6
0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 1, 0, 1, 0, 0, 1, 2, 0, 2, 1, 2, 0, 1, 0, 2, 0, 1, 1, 2, 0, 3, 1, 3, 0, 2, 0, 3, 0, 1, 1, 3, 0, 3, 0, 2, 0, 1, 0, 3, 0, 1, 1, 2, 0, 3, 1, 4, 0, 2, 0, 4, 0, 1, 0, 1, 0, 4, 1, 3, 0, 2, 0, 3, 0, 1, 1, 3, 0, 4, 1, 4, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,17

LINKS

Table of n, a(n) for n=1..90.

R. Zumkeller, Integer-sided triangles

FORMULA

a(n) = A070088(n) - A070090(n).

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * ([i = k] + [i = n-i-k] - [k = n-i-k]) * A010051(i) * A010051(k) * A010051(n-i-k), where [] is the Iverson bracket. - Wesley Ivan Hurt, May 14 2019

EXAMPLE

For n=17 there are A005044(17)=8 integer triangles: [1,8,8], [2,7,8], [3,6,8], [3,7,7], [4,5,8], [4,6,7], [5,5,7] and [5,6,6]: four are isosceles: [1<8=8], [3<7=7], [5=5<7] and [5<6=6], but only two of them consist of primes, therefore a(17)=2.

MATHEMATICA

Table[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[n - i - k] - PrimePi[n - i - k - 1]) (KroneckerDelta[i, k] + KroneckerDelta[i, n - i - k] - KroneckerDelta[k, n - i - k]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] (* Wesley Ivan Hurt, May 14 2019 *)

CROSSREFS

Cf. A070080, A070081, A070082, A059169, A070100, A070108, A070117.

Sequence in context: A115525 A241910 A065717 * A307837 A123671 A191261

Adjacent sequences:  A070089 A070090 A070091 * A070093 A070094 A070095

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, May 05 2002

STATUS

approved

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Last modified October 22 13:19 EDT 2019. Contains 328318 sequences. (Running on oeis4.)