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

1,1

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 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.

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

Cf. A000040, A000567, A001222, A014612.

Sequence in context: A032934 A370977 A337824 * A276114 A022117 A042571

Adjacent sequences: A114603 A114604 A114605 * A114607 A114608 A114609

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 17 2006

Status

approved