A114606 Numbers k such that the k-th octagonal number is 3-almost prime.

2, 15, 17, 19, 21, 25, 29, 31, 33, 35, 41, 51, 55, 65, 73, 77, 79, 83, 89, 91, 93, 95, 97, 101, 107, 111, 123, 131, 133, 139, 141, 145, 149, 151, 155, 157, 173, 179, 183, 197, 201, 203, 205, 215, 221, 223, 227, 229, 233, 237, 241, 247, 253

Offset: 1

Author

Jonathan Vos Post, Feb 17 2006

Keywords

• easy
• nonn

Comments

It is necessary but not sufficient that k must be either prime or semiprime.

Examples

			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].
		

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, A014612.

Programs

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 *)

Formula

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

Numbers k such that A000567(k) is a term of A014612.

Numbers k such that A001222(A000567(k)) = 3.

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

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.