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

1,1

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.

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.

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.

Example

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.

Mathematica

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

Crossrefs

Cf. A000384, A001222, A046306.

Sequence in context: A295155 A118146 A349210 * A227548 A282112 A172468

Adjacent sequences: A114501 A114502 A114503 * A114505 A114506 A114507

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 14 2006

Extensions

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

Status

approved