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

1,1

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.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..5000

Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A187218 A239253 A061118 * A069064 A024410 A200521

Adjacent sequences: A097819 A097820 A097821 * A097823 A097824 A097825

Keyword

nonn

Author

Hugo Pfoertner, Aug 26 2004

Extensions

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

Status

approved