A114456 Numbers k such that the k-th hexagonal number is a 5-almost prime.

8, 14, 16, 18, 20, 24, 28, 36, 38, 40, 41, 44, 54, 74, 77, 78, 84, 86, 90, 92, 100, 102, 105, 110, 113, 123, 124, 125, 126, 130, 132, 135, 136, 143, 148, 149, 153, 156, 164, 165, 170, 171, 184, 185, 186, 194, 207, 210, 213, 215, 218, 220, 225, 232, 234, 236

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) = 8 because HexagonalNumber(8) = H(8) = 8*(2*8-1) = 120 = 2^3 * 3 * 5 is a 5-almost prime.
a(2) = 14 because H(14) = 14*(2*14-1) = 378 = 2 * 3^3 * 7 is a 5-almost prime.
a(3) = 18 because H(18) = 18*(2*18-1) = 630 = 2 * 3^2 * 5 * 7 is a 5-almost prime.
a(20) = 100 because H(100) = 100*(2*100-1) = 19900 = 2^2 * 5^2 * 199 is a 5-almost prime.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
• Eric Weisstein's World of Mathematics, Almost Prime.
• Eric Weisstein's World of Mathematics, Hexagonal Number.

Crossrefs

Cf. A000384, A001222, A014614.

Programs

Mathematica

Select[Range[300], PrimeOmega[#*(2*# - 1)] == 5 &] (* Giovanni Resta, Jun 14 2016 *)
Select[Range[300],PrimeOmega[PolygonalNumber[6,#]]==5&] (* Harvey P. Dale, Jan 15 2023 *)

Formula

Numbers k such that hexagonal number A000384(k) is an element of A014614.

Numbers k such that A001222(A000384(k)) = 5.

Numbers k such that A001222(k*(2*k-1)) = 5.

Extensions

Missing a(3)=16 and more terms from Giovanni Resta, Jun 14 2016