[seqfan] Primes on the right line on the plane "n - prime(n)"

[zak seidov]

20 primes on the right line on the plane "n - prime(n)":  y=197+6*(x-45).

{n,prime(n)}:
{{45, 197}, {51, 233}, {52, 239}, {54, 251}, {55, 257}, {56, 263}, {57, 269}, {64, 311}, {71, 353}, {72, 359}, {76, 383}, {77, 389}, {79, 401}, {86, 443}, {87, 449}, {89, 461}, {92, 479}, {94, 491}, {96, 503}, {97, 509}}.

Worth submitting(?):
1. Indices of primes: 
{45,51,52,54,55,56,57,64,71,72,76,77,79,86,87,89,92,94,96,97}
2. Primes: 
{197,233,239,251,257,263,269,311,353,359,383,389,401,443,449,461,479,491,503,509} 
3. Indices and primes:  
{{45, 197}, {51, 233}, {52, 239}, {54, 251}, {55, 257}, {56, 263}, {57, 269}, {64, 311}, {71, 353}, {72, 359}, {76, 383}, {77, 389}, {79, 401}, {86, 443}, {87, 449}, {89, 461}, {92, 479}, {94, 491}, {96, 503}, {97, 509}}
Are there cases with  more primes on the right line?
Thx, Zak