A104012 Indices of centered dodecahedral numbers (A005904) which are semiprimes (A001358).

1, 2, 3, 5, 6, 11, 14, 15, 21, 26, 30, 35, 36, 44, 54, 63, 69, 74, 81, 114, 128, 131, 135, 138, 153, 165, 168, 191, 194, 209, 216, 224, 228, 231, 239, 261, 270, 303, 315, 321, 323, 326, 330, 336, 345, 363, 380, 384, 398, 404, 410, 411, 414, 429, 440, 443, 455, 468, 470

Offset: 1

Author

Jonathan Vos Post, Feb 24 2005

Keywords

• easy
• nonn

Comments

Because the cubic polynomial for centered dodecahedral numbers factors into n time an irreducible quadratic, the dodecahedral numbers can never be prime, but can be semiprime iff (2*n+1) is prime and (5*n^2+5*n+1) is prime. Centered dodecahedral numbers (A005904) are not to be confused with dodecahedral numbers (A006566) = n(3n-1)(3n-2)/2, nor with rhombic dodecahedral numbers (A005917).

Intersection of A005097 and A090563. - Michel Marcus, Apr 30 2016

Examples

			a(1) = 1 because A005904(1) = 33 = 3 * 11, which is semiprime.
a(2) = 2 because A005904(2) = 155 = 5 * 31, which is semiprime.
a(3) = 3 because A005904(3) = 427 = 7 * 61, which is semiprime.
a(4) = 5 because A005904(5) = 1661 = 11 * 151.
194 is in this sequence because A005904(194) = 73579739 = 389 * 189151, which is semiprime.
		

Links

• Zak Seidov, Table of n, a(n) for n = 1..1000
• B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985),4545-4558.

Crossrefs

Cf. A001222, A001358, A005904, A090563, A104011.

Programs

PARI

isok(n) = isprime(2*n+1) && isprime(5*n^2+5*n+1); \\ Michel Marcus, Apr 30 2016

Formula

n such that A001222(A005904(n)) = 2. n such that Bigomega((2*n+1)*(5*n^2 + 5*n + 1)) is 2. n such that A104011(n) = 2.