A102853 Number of prime factors (with multiplicity) of number of points on surface of square pyramid.

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, 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, 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

Offset

0,3

Comments

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,...

References

H. S. M. Coxeter, "Polyhedral numbers," in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.

B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.

A. F. Wells, Three-Dimensional Nets and Polyhedra, Fig. 15.1 (e).

Links

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cino Hilliard, 3n^2+2 = x^2 is impossible for all integers x

Eric Weisstein's World of Mathematics, Prime Factor.

Formula

a(n) = A001222(A005918(n)). a(n) = Bigomega(A005918(n)). a(n) = A001222(3*n^2 + 2).

Example

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).
a(197) = 2 because A005918(197) = 3*197^2+2 = 116429 = 173 * 673.
a(223) = 2 because A005918(223) = 3*223^2+2 = 149189 = 193 * 773.
a(265) = 2 because A005918(265) = 3*265^2+2 = 210677 = 457 * 461.
a(105) = 3 because A005918(105) = 3*105^2+2 = 33077 = 11 * 31 * 97.
a(127) = 3 because A005918(127) = 3*127^2+2 = 48389 = 11 * 53 * 83.

Prog

(PARI) a(n)=bigomega(3*n^2+2) \\ Charles R Greathouse IV, Feb 05 2017

Crossrefs

Cf. A001222, A005918.

Sequence in context: A275832 A237839 A101608 * A304099 A293390 A363654

Adjacent sequences: A102850 A102851 A102852 * A102854 A102855 A102856

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 28 2005

Extensions

a(0) corrected by Charles R Greathouse IV, Feb 05 2017

Status

approved