A114505 Numbers k such that the k-th hexagonal number is a 7-almost prime.

48, 64, 68, 72, 80, 88, 96, 104, 108, 122, 140, 162, 168, 188, 203, 208, 216, 228, 230, 240, 243, 264, 272, 280, 308, 312, 324, 360, 378, 380, 396, 408, 410, 424, 428, 438, 440, 446, 450, 473, 486, 513, 518, 527, 544, 564, 567, 572, 578, 620, 638, 662, 666, 675, 689, 696

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) = 48 because HexagonalNumber(48) = H(48) = 48*(2*48-1) = 4560 = 2^4 * 3 * 5 * 19 is a 7-almost prime.
a(2) = 64 because H(64) = 64*(2*64-1) = 8128 = 2^6 * 127 is a 7-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, A046308.

Programs

Mathematica

Select[Range[800],PrimeOmega[#(2#-1)]==7&] (* Harvey P. Dale, Jul 20 2013 *)
Position[PrimeOmega[PolygonalNumber[6,Range[700]]],7]//Flatten (* Harvey P. Dale, Jan 10 2024 *)

Formula

Numbers k such that hexagonal number A000384(k) is an element of A046308.

Numbers k such that A001222(A000384(k)) = 7.

Numbers k such that A001222(k*(2*k-1)) = 7.