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

%I A097822 #23 Feb 16 2025 08:32:54
%S A097822 420,431,491,492,514,533,573,574,603,614,655,686,738,775,798,858,861,
%T A097822 890,895,901,904,917,919,942,984,989,1025,1059,1116,1130,1162,1169,
%U A097822 1188,1215,1222,1224,1245,1251,1253,1268,1271,1318,1321,1334,1365,1374,1407
%N A097822 Numbers n such that n^2+n+41 (Euler's "prime generating polynomial") has more than 2 prime factors.
%C A097822 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.
%H A097822 Harvey P. Dale, <a href="/A097822/b097822.txt">Table of n, a(n) for n = 1..5000</a>
%H A097822 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>
%e A097822 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.
%t A097822 Select[Range[1500],PrimeOmega[#^2+#+41]>2&] (* _Harvey P. Dale_, Dec 26 2017 *)
%o A097822 (PARI) isok(n) = #factor(n^2+n+41)~ > 2; \\ _Michel Marcus_, Sep 07 2017
%Y A097822 Cf. A002837, A007634, A005846, A097823, A145293.
%K A097822 nonn
%O A097822 1,1
%A A097822 _Hugo Pfoertner_, Aug 26 2004
%E A097822 Corrected a(19) by _Hugo Pfoertner_, Sep 07 2017