A114621 Numbers k such that the k-th octagonal number is 5-almost prime.

8, 10, 12, 20, 26, 28, 45, 58, 63, 68, 76, 81, 82, 92, 99, 106, 115, 116, 129, 146, 159, 165, 171, 172, 188, 195, 202, 212, 213, 218, 225, 236, 255, 259, 261, 268, 273, 279, 298, 309, 325, 339, 343, 351, 362, 375, 387, 395, 399

Offset: 1

Author

Jonathan Vos Post, Feb 17 2006

Keywords

• easy
• nonn

Comments

It is necessary but not sufficient that k must be prime (A000040), semiprime (A001358), 3-almost prime (A014612), or 4-almost prime (A014613).

Examples

			a(1) = 8 because OctagonalNumber(8) = Oct(8) = 8*(3*8-2) = 176 = 2^4 * 11 has exactly 5 prime factors (four are all equally 2; factors need not be distinct). Also, 176 = Oct(8) = Oct(Oct(2)), an iterated octagonal number. Also, 176 is a pentagonal number, hence a term of A046189 octagonal pentagonal numbers.
a(2) = 10 because Oct(10) = 10*(3*10-2) = 280 = 2^3 * 5 * 7 is 5-almost prime.
a(4) = 20 because Oct(20) = 20*(3*20-2) = 1160 = 2^3 * 5 * 29.
a(5) = 26 because Oct(26) = 26*(3*26-2) = 1976 = 2^3 * 13 * 19.
a(19) = 129 because Oct(129) = 129*(3*129-2) = 49665 = 3 * 5 * 7 * 11 * 43 is 5-almost prime (in this case, the 5 prime factors are distinct).
		

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, Octagonal Number.

Crossrefs

Cf. A000040, A000567, A001222, A001358, A014612, A014613, A014614.

Programs

Mathematica

Select[Range[500], PrimeOmega[PolygonalNumber[8, #]] == 5 &] (* Amiram Eldar, Oct 07 2024 *)

Formula

Numbers k such that k*(3*k-2) has exactly five prime factors (with multiplicity).

Numbers k such that A000567(k) is a term of A014614.

Numbers k such that A001222(A000567(k)) = 5.

Numbers k such that A001222(k) + A001222(3*k-2) = 5.

Numbers k such that [(3*k-2)*(3*k-1)*(3*k)]/[(3*k-2)+(3*k-1)+(3*k)] is a term of A014614.

Extensions

12, 63, 99 inserted and 117 removed by R. J. Mathar, Dec 22 2010