A114504 Numbers k such that the k-th hexagonal number is a 6-almost prime.

50, 56, 60, 63, 81, 95, 98, 112, 116, 120, 138, 150, 152, 158, 172, 180, 182, 189, 196, 198, 204, 212, 221, 238, 242, 252, 260, 266, 275, 276, 296, 300, 304, 306, 315, 328, 332, 333, 340, 344, 348, 350, 356, 363, 374, 375, 388, 390, 405, 413, 420, 423, 434, 452, 455, 456, 459, 462, 472

Offset: 1

Author

Jonathan Vos Post, Feb 14 2006

Keywords

• easy
• nonn

Comments

There are no prime hexagonal numbers. The k-th hexagonal number A000384(k) = k*(2*k-1) is semiprime iff both k and 2*k-1 are primes iff A000384(k) is an element of A001358 iff k is an element of A005382.

Examples

			a(1) = 50 because HexagonalNumber(50) = H(50) = 50*(2*50-1) = 4950 = 2 * 3^2 * 5^2 * 11 is a 6-almost prime.
a(2) = 56 because H(56) = 56*(2*56-1) = 6216 = 2^3 * 3 * 7 * 37 is a 6-almost prime.
a(5) = 81 because H(81) = 81*(2*81-1) = 13041 = 3^4 * 7 * 23 is a 6-almost prime.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Almost Prime.
• Eric Weisstein's World of Mathematics, Hexagonal Number.

Crossrefs

Cf. A000384, A001222, A046306.

Programs

Mathematica

Select[Range[500], PrimeOmega[PolygonalNumber[6, #]] == 6 &] (* Amiram Eldar, Oct 06 2024 *)

Formula

Numbers k such that hexagonal number A000384(k) is an element of A046306.

Numbers k such that A001222(A000384(k)) = 6.

Numbers k such that A001222(k*(2*k-1)) = 6.

Extensions

199 replaced by 198 by R. J. Mathar, Dec 22 2010