A097822 Numbers n such that n^2+n+41 (Euler's "prime generating polynomial") has more than 2 prime factors.

420, 431, 491, 492, 514, 533, 573, 574, 603, 614, 655, 686, 738, 775, 798, 858, 861, 890, 895, 901, 904, 917, 919, 942, 984, 989, 1025, 1059, 1116, 1130, 1162, 1169, 1188, 1215, 1222, 1224, 1245, 1251, 1253, 1268, 1271, 1318, 1321, 1334, 1365, 1374, 1407

Offset: 1

Author

Hugo Pfoertner, Aug 26 2004

Keywords

• nonn

Comments

All visible sequence terms give exactly 3 prime factors. The smallest composite of the form p(n)=n^2+n+41 with 4 prime factors occurs for p(1721)=2963603=43*41^3. Smallest n with 4 distinct prime factors: p(2911)=8476873=83*53*47*41, smallest n with 5 prime factors: p(14144)=200066921=47^4*41, smallest n with 5 distinct prime factors: p(38913)=1514260523=173*71*61*47*43.

Examples

			a(1)=420 because 420^2+420+41=176861=71*53*47 is the first n for which p(n)=n^2+n+41 has more than 2 prime factors. For all smaller n p(n) is either prime or semiprime.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..5000
• Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Crossrefs

Cf. A002837, A007634, A005846, A097823, A145293.

Programs

Mathematica

Select[Range[1500],PrimeOmega[#^2+#+41]>2&] (* Harvey P. Dale, Dec 26 2017 *)

PARI

isok(n) = #factor(n^2+n+41)~ > 2; \\ Michel Marcus, Sep 07 2017

Extensions

Corrected a(19) by Hugo Pfoertner, Sep 07 2017