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A102294
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Number of prime divisors (with multiplicity) of icosahedral numbers.
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1
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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
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1,2
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COMMENTS
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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, ...
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LINKS
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Table of n, a(n) for n=1..100.
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FORMULA
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a(n) = A001222(A006564(n)). Bigomega(n*(5*n^2 - 5*n + 2)/2).
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EXAMPLE
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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.
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MATHEMATICA
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Table[PrimeOmega[n*(5*n^2-5*n+2)/2], {n, 120}] (* Harvey P. Dale, Jun 06 2015 *)
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PROG
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(PARI) a(n)=bigomega(n)+bigomega(5*binomial(n, 2)+1) \\ Charles R Greathouse IV, Mar 09, 2012
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CROSSREFS
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Cf. A001222, A006564.
Sequence in context: A284721 A126659 A246917 * A021287 A124887 A304903
Adjacent sequences: A102291 A102292 A102293 * A102295 A102296 A102297
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KEYWORD
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easy,nonn
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AUTHOR
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Jonathan Vos Post, Feb 19 2005
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STATUS
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approved
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