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Number of distinct prime factors of n^3 + n^2 + 71.
2

%I #21 Jun 03 2019 03:18:33

%S 1,1,1,1,1,2,2,1,1,1,1,1,2,1,1,1,1,1,2,2,2,2,2,2,2,2,2,1,1,1,2,2,1,1,

%T 1,1,1,2,2,2,2,2,2,2,3,2,1,1,1,1,3,2,2,2,2,1,3,3,2,1,2,1,1,3,2,3,1,2,

%U 1,1,3,2,1,2,3,2,2,2,1,3,1,1,3,2,1,2

%N Number of distinct prime factors of n^3 + n^2 + 71.

%C This cubic equation with small positive coefficients is strangely rich in primes and semiprimes. The first 44 consecutive values, for n = 0, 1, 2, ..., 43, are all either prime (23 of them) or semiprime (21 of them), before the first 3-almost prime value is encountered.

%H T. D. Noe, <a href="/A105551/b105551.txt">Table of n, a(n) for n = 0..1000</a>

%H Ulrich Abel and Hartmut Siebert, <a href="https://www.jstor.org/stable/2323773">Sequences with Large Numbers of Prime Values</a>, Am. Math. Monthly 100, 167-169, 1993.

%H Robin Forman, <a href="https://www.jstor.org/stable/2324063">Sequences with Many Primes</a>, Amer. Math. Monthly 99, 548-557, 1992.

%H Betty Garrison, <a href="https://www.jstor.org/stable/2324515">Polynomials with Large Numbers of Prime Values</a>, Amer. Math. Monthly 97, 316-317, 1990.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>.

%F a(n) = A001221(n^3 + n^2 + 71).

%e a(0) = 1 because 0^3 + 0^2 + 71 = 71 is prime.

%e a(1) = 1 because 1^3 + 1^2 + 71 = 73 is prime.

%e a(2) = 1 because 2^3 + 2^2 + 71 = 83 is prime.

%e a(3) = 1 because 3^3 + 3^2 + 71 = 107 is prime.

%e a(4) = 1 because 3^3 + 3^2 + 71 = 151 is prime.

%e a(5) = 2 because 3^3 + 3^2 + 71 = 221 = 13 * 17 is the first semiprime.

%e a(44) = 3 because 44^3 + 44^2 + 71 = 87191 = 13 * 19 * 353 is the first 3-almost prime for nonnegative integers n.

%t Table[PrimeNu[n^3+n^2+71],{n,0,90}] (* _Harvey P. Dale_, Oct 09 2012 *)

%o (PARI) a(n)=omega(n^3 + n^2 + 71) \\ _Charles R Greathouse IV_, Jan 31 2017

%Y Cf. A000040, A001358, A005846, A007635, A007641, A048988, A050265, A050268, A050267, A050266.

%K easy,nonn

%O 0,6

%A _Jonathan Vos Post_, May 03 2005

%E More terms from _Robert G. Wilson v_, May 21 2005