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

%I A253827 #18 Feb 16 2025 08:33:24
%S A253827 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,
%T A253827 68,16,71,30,23,37,37,67,44,54,55,54,26,65,50,70,68,79,43,60,70,52,51,
%U A253827 132,38,60,100,59,111,114,84,77,68,78,105,49,67,124,145,35
%N A253827 a(n) is the number of primes of the form x^2 + x + prime(n) for 0 <= x <=prime(n).
%C A253827 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)
%C A253827 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
%H A253827 Michel Lagneau, <a href="/A253827/b253827.txt">Table of n, a(n) for n = 1..4000</a>
%H A253827 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LuckyNumberofEuler.html">Lucky Number of Euler</a>
%e A253827 a(13) = 40 because prime(13) = 41 and x^2 + x + 41 generates 40 prime numbers for x = 0..41.
%p A253827 f:= proc(n)
%p A253827 local p,x;
%p A253827 p:= ithprime(n);
%p A253827 nops(select(isprime, [seq(x^2+x+p,x=0..p)]))
%p A253827 end proc:
%p A253827 seq(f(n), n=1..100); # _Robert Israel_, Jan 16 2015
%t A253827 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
%t A253827 Table[With[{p=Prime[n]},Count[Table[x^2+x+p,{x,0,p}],_?PrimeQ]],{n,70}] (* _Harvey P. Dale_, May 27 2018 *)
%o A253827 (PARI) a(n) = my(p=prime(n)); sum(k=0, p, isprime(subst(x^2+x+p, x, k))); \\ _Michel Marcus_, Jan 16 2015
%Y A253827 Cf. A000040, A005846, A014556, A228123.
%K A253827 nonn
%O A253827 1,2
%A A253827 _Michel Lagneau_, Jan 16 2015