A208936 -id:A208936 - cp's OEIS frontend

A208645 Least x>0 such that x^2+x+n is not prime.

2, 4, 1, 2, 1, 4, 1, 1, 1, 2, 1, 10, 1, 1, 1, 2, 1, 16, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 40, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 0

M. F. Hasler, Mar 03 2012

nonn

By definition, a(n)>0 for all n, and a(n)>1 if n+2 is prime.

			a(0)=2 since 1^2+1+0=2 is prime, but 2^2+2+0=6 is composite.
a(1)=4 since 1^2+1+1=2, 2^2+2+1=7 and 3^2+3+1=13 are prime, but 4^2+4+1=21 is composite.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A208936, A014556, A092749.

Cf. A117530, A117531.

lx[n_]:=Module[{x=1},While[PrimeQ[x^2+x+n],x++];x]; Array[lx, 90, 0] (* Harvey P. Dale, Aug 14 2013 *)

a(n)=for( x=1, n+3, isprime(x^2+x+n) || return(x))

A371896 a(n) is the length of the uninterrupted sequence of primes generated by the polynomial f(x) = x^2 + x + p for x=0,1,..., where p=A001359(n).

Original entry on oeis.org

2, 4, 10, 16, 2, 40, 2, 2, 4, 3, 2, 2, 2, 3, 2, 4, 2, 2, 2, 3, 5, 2, 2, 3, 2, 2, 2, 2, 5, 2, 2, 3, 2, 3, 3, 2, 2, 2, 2, 4, 2, 2, 7, 2, 3, 2, 5, 2, 4, 4, 6, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 3, 5, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2

Offset: 1

Peter Rowlett, Apr 11 2024

nonn

p=A001359(n) is the smaller prime of a twin prime pair so that f(0) = p and f(1) = p+2 are both primes so a(n) >= 2 and this sequence is the terms >= 2 in A208936.

			For n=6, p = A001359(n) = 41 and f(x) = x^2 + x + 41 is Euler's polynomial which generates primes f(x) for x=0,1,2,...,39, which is 40 terms so a(6) = 40 (cf. A202018).

L. Euler, Nouveaux Mémoires de l'Académie royale des Sciences, 1772, p. 36.

Peter Rowlett, Table of n, a(n) for n = 1..10000

Cf. A001359, A202018, A208936.

A001359[1] = 3;
A001359[n_] := A001359[n] = (p = NextPrime[A001359[n - 1]];
  While[!PrimeQ[p + 2], p = NextPrime[p]]; p)
A371896[n_] := With[{p = A001359@n}, k = 1;
  While[PrimeQ[k^2 + k + p], k++]; k]
(* Leo C. Stein, May 09 2024 *)

cp's OEIS Frontend

A208645 Least x>0 such that x^2+x+n is not prime.

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