A103981 Number of prime factors (with multiplicity) of octahedral numbers (A005900).

0, 0, 2, 1, 3, 2, 2, 3, 4, 2, 3, 5, 4, 2, 3, 3, 7, 2, 4, 2, 5, 2, 4, 2, 4, 4, 4, 3, 4, 4, 3, 2, 6, 2, 4, 4, 4, 3, 5, 3, 6, 3, 3, 4, 4, 3, 4, 3, 6, 3, 4, 4, 5, 2, 5, 3, 7, 3, 3, 3, 5, 3, 4, 4, 7, 5, 3, 3, 4, 3, 8, 2, 5, 4, 4, 3, 4, 4, 4, 4, 7, 5, 3, 3, 5, 3, 3

Offset: 0

Author

Jonathan Vos Post, Feb 24 2005

Keywords

• easy
• nonn

Comments

When a(n) = 2, n is a term of A103982: indices of octahedral numbers (A005900) which are semiprimes.

Examples

			a(3) = 1 because OctahedralNumber(3) = A005900(3) = 19, which is prime and thus has only one prime factor. Because the cubic polynomial for octahedral numbers factors into n time a quadratic, the octahedral numbers can never be prime after a(3) = 19.
a(4) = 3 because A005900(4) = (2*4^3 + 4)/3 = 44 = 2 * 2 * 11, which has (with multiplicity) three prime factors.
		

References

• J. H. Conway and R. K. Guy, The Book of Numbers, New York, Springer-Verlag, p. 50, 1996.
• L. E. Dickson, History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1952.

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.
• Eric Weisstein's World of Mathematics, Octahedral Number.

Crossrefs

Cf. A001222, A005900, A103946, A103982.

Programs

Maple

seq(numtheory:-bigomega((2*n^3+n)/3),n=0..100); # Robert Israel, Aug 10 2014

Mathematica

a[n_] := PrimeOmega[n*(2*n^2 + 1)/3]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Oct 11 2024 *)

Formula

a(n) = A001222(A005900(n)), n>0. a(n) = Bigomega((2*n^3 + n)/3), n>0.

Extensions

More terms from Wesley Ivan Hurt, Aug 11 2014