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Numbers k such that the k-th octagonal number is 5-almost prime.
2

%I #18 Feb 16 2025 08:32:59

%S 8,10,12,20,26,28,45,58,63,68,76,81,82,92,99,106,115,116,129,146,159,

%T 165,171,172,188,195,202,212,213,218,225,236,255,259,261,268,273,279,

%U 298,309,325,339,343,351,362,375,387,395,399

%N Numbers k such that the k-th octagonal number is 5-almost prime.

%C It is necessary but not sufficient that k must be prime (A000040), semiprime (A001358), 3-almost prime (A014612), or 4-almost prime (A014613).

%H Amiram Eldar, <a href="/A114621/b114621.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OctagonalNumber.html">Octagonal Number</a>.

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

%F Numbers k such that A000567(k) is a term of A014614.

%F Numbers k such that A001222(A000567(k)) = 5.

%F Numbers k such that A001222(k) + A001222(3*k-2) = 5.

%F Numbers k such that [(3*k-2)*(3*k-1)*(3*k)]/[(3*k-2)+(3*k-1)+(3*k)] is a term of A014614.

%e a(1) = 8 because OctagonalNumber(8) = Oct(8) = 8*(3*8-2) = 176 = 2^4 * 11 has exactly 5 prime factors (four are all equally 2; factors need not be distinct). Also, 176 = Oct(8) = Oct(Oct(2)), an iterated octagonal number. Also, 176 is a pentagonal number, hence a term of A046189 octagonal pentagonal numbers.

%e a(2) = 10 because Oct(10) = 10*(3*10-2) = 280 = 2^3 * 5 * 7 is 5-almost prime.

%e a(4) = 20 because Oct(20) = 20*(3*20-2) = 1160 = 2^3 * 5 * 29.

%e a(5) = 26 because Oct(26) = 26*(3*26-2) = 1976 = 2^3 * 13 * 19.

%e a(19) = 129 because Oct(129) = 129*(3*129-2) = 49665 = 3 * 5 * 7 * 11 * 43 is 5-almost prime (in this case, the 5 prime factors are distinct).

%t Select[Range[500], PrimeOmega[PolygonalNumber[8, #]] == 5 &] (* _Amiram Eldar_, Oct 07 2024 *)

%Y Cf. A000040, A000567, A001222, A001358, A014612, A014613, A014614.

%K easy,nonn,changed

%O 1,1

%A _Jonathan Vos Post_, Feb 17 2006

%E 12, 63, 99 inserted and 117 removed by _R. J. Mathar_, Dec 22 2010