A339415 Table read by rows. If p=A098058(n+1), q is the next prime after p, and r=(p+q)/2, row n consists of the areas (in increasing order) of triangles with vertices (p,p), (s,r-s), (q,q), where s and r-s are prime.
0, 0, 2, 4, 8, 0, 36, 60, 4, 8, 16, 0, 36, 72, 84, 4, 16, 20, 32, 36, 72, 108, 132, 54, 90, 150, 2, 14, 22, 26, 34, 46, 54, 90, 126, 162, 174, 10, 14, 34, 46, 50, 62, 54, 90, 126, 198, 210, 0, 144, 180, 216, 240, 16, 20, 40, 44, 56, 64, 76, 92, 14, 26, 34, 50, 70, 86, 94, 98, 14, 98, 182, 266
Offset: 1
Examples
With p=A098058(5)=17, q=19, r=18, the values of s are 5, 7, 11, 13, corresponding to areas 4, 8, 8, 4 respectively, so row 4 is (4,8). The first 10 rows are 0 0 2 4, 8 0, 36, 60 4, 8, 16 0, 36, 72, 84 4, 16, 20, 32 36, 72, 108, 132 54, 90, 150
Links
- Robert Israel, Table of n, a(n) for n = 1..10020 (rows 1 to 261, flattened)
Programs
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Maple
R:= 0: count:= 1: q:= 5: nrows:= 1: printf("0\n"): while nrows < 20 do p:= q; q:= nextprime(q); if p+q mod 4 <> 0 then next fi; nrows:= nrows+1; r:= (p+q)/2; T:= select(t -> isprime(t) and isprime(r-t), [$ceil(r/2)..r]); count:= count + nops(T); V:= map(t -> abs((p-q)*(p+q-4*t)/4), T); R:= R, op(V); printf("%a\n",V); od:
Formula
The area of the triangle with vertices (p,p), (s,r-s), (q,q) is (q-p)*|p+q-4*s|/4.
Comments