A253827 a(n) is the number of primes of the form x^2 + x + prime(n) for 0 <= x <=prime(n).

1, 2, 4, 4, 10, 4, 16, 6, 10, 13, 14, 16, 40, 8, 26, 19, 34, 21, 36, 28, 18, 18, 34, 27, 31, 68, 16, 71, 30, 23, 37, 37, 67, 44, 54, 55, 54, 26, 65, 50, 70, 68, 79, 43, 60, 70, 52, 51, 132, 38, 60, 100, 59, 111, 114, 84, 77, 68, 78, 105, 49, 67, 124, 145, 35

Offset: 1

Author

Michel Lagneau, Jan 16 2015

Keywords

• nonn

Comments

Equivalently, number of distinct primes of the form x^2 - x + prime(n) for 0 <= x <= prime(n). (The point is that x^2 + x = (x+1)^2 - (x+1), so the two forms give the same numbers. x^2 - x + prime(n) is the same for x=0 and x=1, which is why the "distinct" in the comment. - Robert Israel, Oct 09 2016)

1 <= a(n) <= prime(n)-1. a(n) = prime(n)-1 iff n is in A014556. Are there any n > 1 such that a(n) = 1? - Robert Israel, Jan 16 2015

Examples

			a(13) = 40 because prime(13) = 41 and x^2 + x + 41 generates 40 prime numbers for x = 0..41.
		

Links

• Michel Lagneau, Table of n, a(n) for n = 1..4000
• Eric Weisstein's World of Mathematics, Lucky Number of Euler

Crossrefs

Cf. A000040, A005846, A014556, A228123.

Programs

Maple

f:= proc(n)
local p,x;
p:= ithprime(n);
nops(select(isprime, [seq(x^2+x+p,x=0..p)]))
end proc:
seq(f(n), n=1..100); # Robert Israel, Jan 16 2015

Mathematica

lst={};Do[p=Prime[n];k=0;Do[If[PrimeQ[x^2+x+p],k=k+1],{x,0,p}];AppendTo[lst,k],{n,1,100}];lst
Table[With[{p=Prime[n]},Count[Table[x^2+x+p,{x,0,p}],?PrimeQ]],{n,70}] (* _Harvey P. Dale, May 27 2018 *)

PARI

a(n) = my(p=prime(n)); sum(k=0, p, isprime(subst(x^2+x+p, x, k))); \\ Michel Marcus, Jan 16 2015