A114828 Numbers k such that the k-th octagonal number has 9 prime factors counted with multiplicity.

64, 96, 128, 144, 162, 182, 198, 216, 224, 234, 246, 270, 278, 288, 304, 310, 320, 324, 352, 390, 414, 416, 432, 438, 480, 504, 528, 544, 550, 558, 584, 594, 600, 646, 648, 654, 662, 684, 694, 702, 710, 729, 750, 752, 756, 798, 810, 834, 850, 870, 888, 900

Offset: 1

Author

Jonathan Vos Post, Feb 19 2006

Keywords

• easy
• nonn

Comments

k has at most 8 prime factors counted with multiplicity.

Examples

			a(1) = 64 because OctagonalNumber(64) = Oct(64) = 64*(3*64-2) = 12160 = 2^7 * 5 * 19 has exactly 9 prime factors (seven are all equally 2; factors need not be distinct).
a(2) = 96 because Oct(96) = 96*(3*96-2) = 27456 = 2^6 * 3 * 11 * 13 is 9-almost prime [also 27456 = Oct(96) = Oct(Oct(6)) is an iterated octagonal number].
a(3) = 128 because Oct(128) = 128*(3*128-2) = 48896 = 2^8 * 191.
		

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, A046306, A046308, A046310, A046312, A088878, A114606, A114618, A114621, A114634, A114635, A114636.

Programs

Magma

A000567:=func< n | n*(3*n-2) >; Is9almostprime:=func< n | &+[k[2]: k in Factorization(n)] eq 9 >; [ n: n in [2..1000] | Is9almostprime(A000567(n)) ]; // Klaus Brockhaus, Dec 22 2010

Mathematica

Select[Range[900],PrimeOmega[PolygonalNumber[8,#]]==9&] (* James C. McMahon, Jul 30 2024 *)

PARI

isok(k) = bigomega(k*(3*k-2)) == 9; \\ Michel Marcus, Aug 02 2024

Formula

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

Integers k such that A000567(k) is a term of A046312.

Integers k such that A001222(A000567(k)) = 9.

Integers k such that A001222(k) + A001222(3*k-2) = 9.

Integers k such that (3*k-2)*(3*k-1)*(3*k)/((3*k-2)+(3*k-1)+(3*k)) is in A046310.

Extensions

Missing terms inserted by R. J. Mathar, Dec 22 2010

a(40)-a(52) from James C. McMahon, Jul 30 2024

Name edited by David A. Corneth, Jul 31 2024