OFFSET
1,1
COMMENTS
k has at most 8 prime factors counted with multiplicity.
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.
FORMULA
Integers k such that k*(3*k-2) has exactly nine prime factors (with multiplicity).
Integers k such that (3*k-2)*(3*k-1)*(3*k)/((3*k-2)+(3*k-1)+(3*k)) is in A046310.
EXAMPLE
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.
MATHEMATICA
Select[Range[900], PrimeOmega[PolygonalNumber[8, #]]==9&] (* James C. McMahon, Jul 30 2024 *)
PROG
(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
(PARI) isok(k) = bigomega(k*(3*k-2)) == 9; \\ Michel Marcus, Aug 02 2024
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Feb 19 2006
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
STATUS
approved
