

A103981


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


2



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
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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, SpringerVerlag, p. 50, 1996
Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1952.


LINKS



FORMULA



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



KEYWORD

easy,nonn


AUTHOR



EXTENSIONS



STATUS

approved



