A102250 Indices of semiprime Haüy rhombic dodecahedral numbers.

2, 3, 4, 6, 12, 15, 16, 22, 34, 36, 51, 66, 87, 99, 100, 106, 117, 139, 141, 159, 166, 169, 174, 177, 180, 192, 201, 205, 232, 274, 282, 307, 337, 339, 342, 367, 370, 372, 379, 381, 411, 412, 429, 430, 432, 439, 444, 454, 460, 471, 477, 507, 510, 517, 555, 577

Offset: 1

Author

Jonathan Vos Post, Feb 18 2005

Keywords

• easy
• nonn

Comments

Because the Haüy rhombic dodecahedral numbers are A046142(n) = (2*n-1)(8*n^2-14*n+7) no Haüy rhombic dodecahedral number can be prime.

Integers n such that both (2*n-1) and (8*n^2-14*n+7) are primes.

Integers n such that (2*n-1)*(8*n^2-14*n+7) is an element in the intersection of A046142 and A001358.

Examples

			a(3) = 4 because the 3rd Haüy rhombic dodecahedral number is A046142(3) = (2*4-1)(8*4^2-14*4+7) = 553 and because 553 = 7 * 79 is a semiprime.
		

References

• R.-J. Haüy, Essai d'une théorie sur la structure des crystaux appliquée à plusieurs genres de substances crystallisées, 1784.
• H. Steinhaus, Mathematical Snapshots, 3rd ed. New York: Dover, pp. 185-186, 1999.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000 which lists Haüy rhombic dodecahedral numbers as "RhoDod(n).
• Eric Weisstein's World of Mathematics, Rhombic Dodecahedron.
• Eric Weisstein's World of Mathematics, Haüy Construction.

Crossrefs

Cf. A001358, A046142, A085601, A102130.

Programs

Magma

[n: n in [0..600] | IsPrime(2*n-1) and IsPrime(8*n^2-14*n+7)]; // Vincenzo Librandi, Sep 22 2012

Mathematica

Select[ Range[1000], PrimeQ[2# - 1] && PrimeQ[8#^2 - 14# + 7] &]
Select[Range[1000],AllTrue[{2#-1,8#^2-14#+7},PrimeQ]&] (* Harvey P. Dale, Apr 13 2025 *)