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Number of prime factors (with multiplicity) of centered dodecahedral numbers (A005904).
2

%I #14 Oct 11 2024 13:15:27

%S 0,2,2,2,3,2,2,3,3,3,4,2,4,4,2,2,3,3,3,3,3,2,4,3,3,3,2,4,4,3,2,6,3,3,

%T 4,2,2,5,3,3,6,3,4,3,2,4,4,4,3,4,3,3,4,3,2,3,3,4,5,4,3,3,4,2,5,3,3,7,

%U 3,2,3,3,4,4,2,3,5,4,3,3,3,2,4,3,4,4,4,4,3,4,3,4,4,3,5,3,3,6,3,3

%N Number of prime factors (with multiplicity) of centered dodecahedral numbers (A005904).

%C When a(n) = 2, n is a term of A104012: indices of centered dodecahedral numbers (A005904) which are semiprimes.

%H Amiram Eldar, <a href="/A104011/b104011.txt">Table of n, a(n) for n = 0..10000</a>

%H Boon K. Teo and N. J. A. Sloane, <a href="https://doi.org/10.1021/ic00220a025">Magic numbers in polygonal and polyhedral clusters</a>, Inorgan. Chem. 24 (1985), 4545-4558; <a href="http://neilsloane.com/doc/Me117.pdf">author's copy</a>.

%F a(n) = A001222(A005904(n)).

%F a(n) = Bigomega((2*n+1)*(5*n^2 + 5*n + 1)).

%e a(9) = 3 because A005904(9) = 8569 = 11 * 19 * 41, which has 3 prime factors (which happen to have the same number of digits).

%e a(18) = 3 because A005904(18) = 63307 = 29 * 37 * 59.

%e a(96) = 3 because A005904(96) = 8986273 = 101 * 193 * 461.

%e a(126) = 5 because A005904(126) = 20242783 = 11 * 23 * 29 * 31 * 89, which has 5 prime factors (which happen to have the same number of digits).

%t PrimeOmega[(2*n+1)*(5*n^2+5*n+1)] /. n -> Range[0, 99] (* _Giovanni Resta_, Jun 17 2016 *)

%Y Cf. A001222, A005904, A104012.

%K easy,nonn

%O 0,2

%A _Jonathan Vos Post_, Feb 24 2005

%E A missing term inserted by _Giovanni Resta_, Jun 17 2016