A104012 - OEIS

%I #13 Apr 30 2016 03:39:05

%S 1,2,3,5,6,11,14,15,21,26,30,35,36,44,54,63,69,74,81,114,128,131,135,

%T 138,153,165,168,191,194,209,216,224,228,231,239,261,270,303,315,321,

%U 323,326,330,336,345,363,380,384,398,404,410,411,414,429,440,443,455,468,470

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

%C 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).

%C Intersection of A005097 and A090563. - _Michel Marcus_, Apr 30 2016

%H Zak Seidov, <a href="/A104012/b104012.txt">Table of n, a(n) for n = 1..1000</a>

%H B. K. Teo and N. J. A. Sloane, <a href="http://neilsloane.com/doc/magic1/magic1.html">Magic numbers in polygonal and polyhedral clusters</a>, Inorgan. Chem. 24 (1985),4545-4558.

%F 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.

%e a(1) = 1 because A005904(1) = 33 = 3 * 11, which is semiprime.

%e a(2) = 2 because A005904(2) = 155 = 5 * 31, which is semiprime.

%e a(3) = 3 because A005904(3) = 427 = 7 * 61, which is semiprime.

%e a(4) = 5 because A005904(5) = 1661 = 11 * 151.

%e 194 is in this sequence because A005904(194) = 73579739 = 389 * 189151, which is semiprime.

%o (PARI) isok(n) = isprime(2*n+1) && isprime(5*n^2+5*n+1); \\ _Michel Marcus_, Apr 30 2016

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

%K easy,nonn

%O 1,2

%A _Jonathan Vos Post_, Feb 24 2005