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

0,3

Comments

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

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York, Springer-Verlag, p. 50, 1996

Dickson, L. E. 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

Formula

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

Example

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.

Maple

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

Crossrefs

Cf. A001222, A005900, A103946, A103982.

Sequence in context: A336037 A058773 A122805 * A029270 A364503 A090350

Adjacent sequences: A103978 A103979 A103980 * A103982 A103983 A103984

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 24 2005

Extensions

More terms from Wesley Ivan Hurt, Aug 11 2014

Status

approved