A308451 - OEIS

%I #12 Jun 16 2020 13:12:23

%S 0,0,0,0,1,1,2,1,1,0,3,2,2,1,5,3,3,2,2,1,1,0,6,5,5,4,11,9,9,7,7,5,5,3,

%T 12,10,10,8,18,15,15,13,13,11,11,9,21,18,18,15,15,12,12,10,10,8,8,6,

%U 21,19,19,17,33,30,30,27,27,24,24,21,21,19,19,17

%N Number of integer-sided triangles with perimeter n whose largest side length is prime.

%H Robert Israel, <a href="/A308451/b308451.txt">Table of n, a(n) for n = 1..10000</a>

%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))) * c(n-i-k), where c(n) is the prime characteristic (A010051).

%p f:= proc(n) local p,v;

%p   v:= add(1/2*(3*p-n+1)+`if`((n-p)::even,1/2,0), p = select(isprime, [$ceil(n/3)..floor((n-1)/2)]));

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, May 28 2019

%t Table[Sum[Sum[ (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}]

%Y Cf. A010051, A308450.

%K nonn

%O 1,7

%A _Wesley Ivan Hurt_, May 27 2019