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%I #7 Feb 05 2017 18:58:15
%S 1,1,2,1,3,2,3,1,2,3,2,2,3,1,3,1,4,2,2,3,2,3,2,2,3,1,4,2,3,3,3,2,3,2,
%T 3,1,3,2,3,3,5,2,2,2,4,2,4,2,2,3,4,3,2,1,6,2,3,1,4,2,3,4,3,1,3,2,3,1,
%U 3,2,2,5,4,2,4,2,3,1,2,4,2,3,4,2,4,2,4,1,2,4,3,2,2,3,4,1,5,1,2,2,3
%N Number of prime factors (with multiplicity) of number of points on surface of square pyramid.
%C Prime for n = 1, 3, 7, 13, 15, 25, 35, 53, 57, 63, 67, 77, 87, 95, 97, 123, 125, 133, 153, 155, 165, 183, 185, 195, 207, 217, 227, 245, 253, 255, 263, 273, 277, 295, ... Semiprime for n = 2, 5, 8, 10, 11, 17, 18, 20, 22, 23, 27, 31, 33, 37, 41, 42, 43, 45, 47, 48, 52, 55, 59, 65, 69, 70, 73, 75, 78, 80, 83, 85, 88, 91, 92, 98, 99, 101, 102, 103, 108, 109, 111, 113, 115, 117, 118, 120, 137, 139, 140, 143, 145, 147, 150, 151, 157, 158, 162, 167, 168, 169, 175, 178, 188, 189, 190, 193, 197, 199, 203, 209, 211, 213, 218, 223, 225, 228, 232, 237, 238, 239, 241, 243, 249, 252, 257, 262, 267, 272, 275, 283, 287, 290, 297, ... 3-almost prime for n = 4, 6, 9, 12, 14, 19, 21, 24, 28, 29, 30, 32, 34, 36, 38, 39, 49, 51, 56, 60, 62, 64, 66, 68, 76, 81, 90, 93, 100,...
%D H. S. M. Coxeter, "Polyhedral numbers," in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
%D B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
%D A. F. Wells, Three-Dimensional Nets and Polyhedra, Fig. 15.1 (e).
%H Charles R Greathouse IV, <a href="/A102853/b102853.txt">Table of n, a(n) for n = 0..10000</a>
%H Cino Hilliard, <a href="http://groups.msn.com/BC2LCC/3n2isnotsquare.msnw">3n^2+2 = x^2 is impossible for all integers x</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeFactor.html">Prime Factor.</a>
%F a(n) = A001222(A005918(n)). a(n) = Bigomega(A005918(n)). a(n) = A001222(3*n^2 + 2).
%e a(147) = 2 because A005918(147) = 3*147^2+2 = 64829 = 241 * 269, which has exactly two prime factors (which happen to be of the same number of digits).
%e a(197) = 2 because A005918(197) = 3*197^2+2 = 116429 = 173 * 673.
%e a(223) = 2 because A005918(223) = 3*223^2+2 = 149189 = 193 * 773.
%e a(265) = 2 because A005918(265) = 3*265^2+2 = 210677 = 457 * 461.
%e a(105) = 3 because A005918(105) = 3*105^2+2 = 33077 = 11 * 31 * 97.
%e a(127) = 3 because A005918(127) = 3*127^2+2 = 48389 = 11 * 53 * 83.
%o (PARI) a(n)=bigomega(3*n^2+2) \\ _Charles R Greathouse IV_, Feb 05 2017
%Y Cf. A001222, A005918.
%K easy,nonn
%O 0,3
%A _Jonathan Vos Post_, Feb 28 2005
%E a(0) corrected by _Charles R Greathouse IV_, Feb 05 2017
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