A114618 Numbers k such that the k-th octagonal number is 4-almost prime.

4, 9, 27, 39, 49, 57, 59, 69, 75, 85, 87, 105, 109, 117, 119, 121, 125, 143, 147, 153, 161, 169, 175, 177, 185, 187, 199, 207, 217, 219, 231, 235, 239, 245, 249, 265, 267, 269, 275, 283, 285, 289, 291, 299, 301, 305, 311, 319, 321, 327, 329, 333, 335, 345, 349, 357, 359, 361, 363, 371, 381, 385

Offset: 1

Author

Jonathan Vos Post, Feb 17 2006

Keywords

• easy
• nonn

Comments

It is necessary but not sufficient that k must be prime (A000040), semiprime (A001358), or 3-almost prime (A014612).

Examples

			a(1) = 4 because OctagonalNumber(4) = Oct(4) = 4*(3*4-2) = 40 = 2^3 * 5 has exactly 4 prime factors (3 are all equally 2; factors need not be distinct).
a(2) = 9 because Oct(9) = 9*(3*9-2) = 225 = 3^2 * 5^2, a 4-almost prime [225 is also a square, hence a square octagonal number A036428, as is Oct(121)].
a(3) = 27 because Oct(27) = 27*(3*27-2) = 2133 = 3^3 * 79.
a(4) = 39 because Oct(39) = 39*(3*39-2) = 4485 = 3 * 5 * 13 * 23 has exactly 4 prime factors, in this case distinct.
a(26) = 187 because Oct(187) = 187*(3*187-2) = 104533 = 11 * 13 * 17 * 43 [a 4-brilliant number, that is with 4 prime factors that are each the same number of digits in length].
		

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, A036428.

Programs

Mathematica

Select[Range[400],PrimeOmega[#(3#-2)]==4&] (* Harvey P. Dale, Sep 07 2011 *)

Formula

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

Numbers k such that A000567(k) is a term of A014613.

Numbers k such that A001222(A000567(k)) = 4.

Numbers k such that A001222(k) + A001222(3*k-2) = 4.

Numbers k such that [(3*k-2)*(3*k-1)*(3*k)]/[(3*k-2)+(3*k-1)+(3*k)] is a term of A014613.

Extensions

265 inserted by R. J. Mathar, Dec 22 2010