A234511 a(n) is the smallest prime(i) such that (prime(i) - prime(j))/(i - j) = prime(n) with i > j.

5, 11, 29, 97, 641, 1373, 2591, 4327, 8009, 19661, 36451, 134581, 38543, 172969, 212777, 268403, 1784171, 860239, 1562053, 6085103, 6958813, 3422971, 5103029, 14723567, 47973451, 38394329, 36271783, 75837497, 59160181, 47326919, 111660697, 369706811, 323627951

Offset: 1

Author

Robin Garcia, Dec 27 2013

Keywords

• nonn

Comments

(i - j) = 2 for all the calculated terms, with the exception of a(1) where (i - j) = 1 and a(6) where (i - j) = 4.

Examples

			a(3) = 29 is the smallest prime (and 10th prime) such that there is a smaller 8th prime: 19 and (29 - 19) / (10 - 8) = 5 is the third prime.
		

Programs

Mathematica

a[1]=5; a[n_] := Catch[Block[{r = Prime@n, i=2, j, p}, While[True, p = Prime[++i]; j = Mod[i, 2]; While[(j += 2) < i, If[p - Prime@j == r*(i-j), Throw@p]]]]] (* Giovanni Resta, Dec 28 2013 *)

PARI

n=16;c=25000;for(b=2,c,forstep(a=b+2,c,2,d=prime(a)-prime(b);e=(a-b);if(d/e==d\e&d/e==prime(n),print([a,b,prime(a),prime(b),d,e,d/e])))) \\ finds a(16) and in general a(n).

PARI

okp(n, p) = {i = primepi(p); forprime (q = 2, p-1, j = primepi(q); if ((p-q)/(i-j) == prime(n), return(1)););}
a(n) = {p = 2; while (! okp(n, p), p = nextprime(p+1)); p;} \\ Michel Marcus, Dec 28 2013

Extensions

a(17)-a(25) from Giovanni Resta, Dec 28 2013

a(26)-a(33) from Donovan Johnson, Jan 01 2014