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
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.
Programs
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Maple
seq(numtheory:-bigomega((2*n^3+n)/3),n=0..100); # Robert Israel, Aug 10 2014
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Mathematica
a[n_] := PrimeOmega[n*(2*n^2 + 1)/3]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Oct 11 2024 *)
Extensions
More terms from Wesley Ivan Hurt, Aug 11 2014
Comments