A105551 Number of distinct prime factors of n^3 + n^2 + 71.

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, 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, 1, 1, 3, 2, 1, 2, 3, 2, 2, 2, 1, 3, 1, 1, 3, 2, 1, 2

Offset: 0

Author

Jonathan Vos Post, May 03 2005

Keywords

• easy
• nonn

Comments

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.

Examples

			a(0) = 1 because 0^3 + 0^2 + 71 = 71 is prime.
a(1) = 1 because 1^3 + 1^2 + 71 = 73 is prime.
a(2) = 1 because 2^3 + 2^2 + 71 = 83 is prime.
a(3) = 1 because 3^3 + 3^2 + 71 = 107 is prime.
a(4) = 1 because 3^3 + 3^2 + 71 = 151 is prime.
a(5) = 2 because 3^3 + 3^2 + 71 = 221 = 13 * 17 is the first semiprime.
a(44) = 3 because 44^3 + 44^2 + 71 = 87191 = 13 * 19 * 353 is the first 3-almost prime for nonnegative integers n.
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Ulrich Abel and Hartmut Siebert, Sequences with Large Numbers of Prime Values, Am. Math. Monthly 100, 167-169, 1993.
• Robin Forman, Sequences with Many Primes, Amer. Math. Monthly 99, 548-557, 1992.
• Betty Garrison, Polynomials with Large Numbers of Prime Values, Amer. Math. Monthly 97, 316-317, 1990.
• Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.

Crossrefs

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

Programs

Mathematica

Table[PrimeNu[n^3+n^2+71],{n,0,90}] (* Harvey P. Dale, Oct 09 2012 *)

PARI

a(n)=omega(n^3 + n^2 + 71) \\ Charles R Greathouse IV, Jan 31 2017

Formula

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

Extensions

More terms from Robert G. Wilson v, May 21 2005