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A306678
Number of distinct triangles with prime sides and largest side = prime(n).
9
1, 3, 4, 6, 7, 11, 13, 18, 21, 22, 29, 30, 37, 46, 53, 56, 60, 71, 75, 87, 101, 105, 118, 124, 123, 139, 157, 173, 193, 209, 186, 207, 219, 244, 241, 264, 277, 291, 318, 329, 344, 371, 373, 405, 433, 465, 447, 440, 474, 511, 545, 563, 597, 602, 623, 645
OFFSET
1,2
FORMULA
a(n) = A306676(n) + A306677(n).
EXAMPLE
For n=1, there is 1 triangle: {2, 2, 2}, with largest side prime(1) = 2.
For n=2, there are 3 triangles: {2, 2, 3}, {2, 3, 3}, {3, 3, 3}, with largest side prime(2) = 3.
For n=4, there are 6 triangles :{2, 7, 7}, {3, 5, 7}, {3, 7, 7}, {5, 5, 7}, {5, 7, 7}, {7, 7, 7}, with largest side prime(4) = 7. Total = 6 = a(4).
For n=5, largest side = prime(n) = 11. Triangles are {{2, 11, 11}, {3, 11, 11}, {5, 7, 11}, {5, 11, 11}, {7, 7, 11}, {7, 11, 11}, {11, 11, 11}}. Total = 7 = a(5).
MAPLE
#nType=1 for acute triangles, nType=2 for obtuse triangles
#nType=0 for both triangles
CountPrimeTriangles := proc (n, nType := 1)
local aa, oo, j, k, sg, a, b, c, tt, lAcute;
aa := {}; oo := {};
a := ithprime(n);
for j from n by -1 to 1 do
b := ithprime(j);
for k from j by -1 to 1 do
c := ithprime(k);
if a < b+c and abs(b-c) < a and b < c+a and abs(c-a) < b and c < a+b and abs(a-b) < c then
lAcute := evalb(0 < b^2+c^2-a^2);
tt := sort([a, b, c]);
if lAcute then aa := {op(aa), tt} else oo := {op(oo), tt} end if
end if
end do
end do;
return sort(`if`(nType = 1, aa, `if`(nType = 2, oo, `union`(aa, oo))))
end proc:
# Alternative:
with(NumberTheory):
A306678:=proc(n)
local a, i, p;
if n=1 then
1
else
a:=0;
p:=ithprime(n);
for i from pi(nextprime((p-1)/2)) to n do
a:=a+i-pi(nextprime(p-ithprime(i)))+1;
od;
return a
fi;
end proc;
seq(A306678(n), n=1..56); # Felix Huber, Apr 19 2025
KEYWORD
nonn
AUTHOR
César Eliud Lozada, Mar 04 2019
STATUS
approved