A164926 Least prime p such that x^2+x+p produces primes for x=0..n-1 and composite for x=n.

2, 3, 107, 5, 347, 1607, 1277, 21557, 51867197, 11, 180078317, 1761702947, 8776320587, 27649987598537, 291598227841757, 17

Offset: 1

Author

T. D. Noe, Sep 01 2009

Keywords

• hard
• nonn
• more

Comments

Other known values: a(16)=17 and a(40)=41 (which is generated by Euler's polynomial, A005846). There are no other terms less than 10^12. All of Euler's Lucky numbers, A014556, are in this sequence. Assuming the prime k-tuples conjecture, Mollin's theorem 2.1 shows this sequence is defined for n>0.

a(21)=234505015943235329417 found by J. Waldvogel and Peter Leikauf. [Jens Kruse Andersen, Sep 09 2009]

Links

• R. A. Goudsmit, Unusual Prime Number Sequences, Nature, Vol. 214 (June 10, 1967), page 1164.
• R. A. Mollin, Prime-Producing Quadratics, The American Mathematical Monthly, Vol. 104, No. 6 (Jun. - Jul., 1997), pp. 529-544.

Crossrefs

Cf. A005846, A014556, A067774, A210360, A210361, A210362, A210363, A210364, A210365, A211236, A211237, A211238, A211239, A211240, A354585.

Programs

Mathematica

PrimeRun[p_Integer] := Module[{k=0}, While[PrimeQ[k^2+k+p], k++ ]; k]; nn=8; t=Table[0,{nn}]; cnt=0; p=1; While[cnt
				

Extensions

a(14) and a(15) from Jens Kruse Andersen, Sep 09 2009