OFFSET
1,1
COMMENTS
It is necessary but not sufficient that k must be either prime or semiprime.
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
Numbers k such that k*(3*k-2) has exactly three prime factors (with multiplicity).
Numbers k such that [(3*k-2)*(3*k-1)*(3*k)]/[(3*k-2)+(3*k-1)+(3*k)] is a term of A014612.
EXAMPLE
a(1) = 2 because OctagonalNumber(2) = Oct(2) = 2*(3*2-2) = 8 = 2^3 has exactly three prime factors (which are all equally 2; factors need not be distinct).
a(2) = 15 because Oct(15) = 15*(3*15-2) = 645 = 3 * 5 * 43, a 3-almost prime.
a(5) = 21 because Oct(21) = 21*(3*21-2) = 1281 = 3 * 7 * 61 [also, 1281 = Oct(21) = Oct(Oct(3)) is an iterated octagonal number].
a(14) = 65 because Oct(65) = 65*(3*65-2) = 12545 = 5 * 13 * 193 [also, 12545 = Oct(65) = Oct(Oct(5)) is an iterated octagonal number].
a(29) = 133 because Oct(133) = 133*(3*133-2) = 52801 = 7 * 19 * 397 [also, 52801 = Oct(133) = Oct(Oct(7)) is an iterated octagonal number].
MAPLE
A000567 := proc(n) n*(3*n-2) ; end: isA014612 := proc(n) RETURN( numtheory[bigomega](n) = 3) ; end: for n from 1 to 1000 do q := A000567(n) ; if isA014612(q) then printf("%d, ", n) ; fi; od: # R. J. Mathar, Jan 27 2009
MATHEMATICA
Select[Range[500], PrimeOmega[PolygonalNumber[8, #]] == 3 &] (* Amiram Eldar, Oct 07 2024 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Feb 17 2006
STATUS
approved