A102294 Number of prime divisors (with multiplicity) of icosahedral numbers.

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

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^2-5*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 * A358790 A021287 A124887

Adjacent sequences: A102291 A102292 A102293 * A102295 A102296 A102297

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 19 2005

Status

approved