

A102294


Number of prime divisors (with multiplicity) of icosahedral numbers.


1



0, 3, 5, 3, 3, 5, 3, 5, 3, 4, 5, 4, 3, 7, 4, 5, 3, 5, 5, 5, 3, 6, 4, 5, 4, 5, 6, 5, 3, 11, 3, 7, 4, 5, 9, 6, 2, 6, 5, 6, 3, 5, 4, 6, 4, 6, 5, 6, 3, 6, 6, 5, 3, 7, 5, 7, 4, 4, 6, 6, 2, 8, 6, 8, 4, 6, 6, 5, 3, 6, 5, 6, 3, 5, 6, 4, 4, 7, 3, 8, 6, 6, 6, 5, 3, 6, 5, 5, 4, 8, 5, 5, 3, 8, 6, 8, 3, 7, 10, 6
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OFFSET

1,2


COMMENTS

Because the cubic factors into n time a quadratic, the icosahedral numbers can never be prime, but can be semiprime (only if n is prime and also n*(5*n^2  5*n + 2)/2 is prime, as with n = 31, 61, ...


LINKS

Table of n, a(n) for n=1..100.


FORMULA

a(n) = A001222(A006564(n)). Bigomega(n*(5*n^2  5*n + 2)/2).


EXAMPLE

IcosahedralNumber(13) = 5083 = 13 * 17 * 23 so Omega(IcosahedralNumber(13)) = 3.
IcosahedralNumber(37) = 123247 = 37 * 3331 so Omega(IcosahedralNumber(37)) = 2, hence the 37th icosahedral number is the smallest to be semiprime.


MATHEMATICA

Table[PrimeOmega[n*(5*n^25*n+2)/2], {n, 120}] (* Harvey P. Dale, Jun 06 2015 *)


PROG

(PARI) a(n)=bigomega(n)+bigomega(5*binomial(n, 2)+1) \\ Charles R Greathouse IV, Mar 09, 2012


CROSSREFS

Cf. A001222, A006564.
Sequence in context: A284721 A126659 A246917 * A021287 A124887 A304903
Adjacent sequences: A102291 A102292 A102293 * A102295 A102296 A102297


KEYWORD

easy,nonn


AUTHOR

Jonathan Vos Post, Feb 19 2005


STATUS

approved



