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Numbers k such that the k-th octagonal number is 8-almost prime.
2

%I #17 Oct 07 2024 01:13:41

%S 22,70,80,84,102,108,118,126,134,160,174,184,200,230,240,250,252,262,

%T 264,272,318,330,334,336,350,368,378,400,408,420,430,434,444,450,454,

%U 459,462,464,484,494,500,502,510,518,520,522,540,560,564,566,570,574,582

%N Numbers k such that the k-th octagonal number is 8-almost prime.

%C It is necessary but not sufficient that k must be prime (A000040), semiprime (A001358), 3-almost prime (A014612), 4-almost prime (A014613), 5-almost prime (A014614), 6-almost prime (A046306), or 7-almost prime (A046308).

%H Amiram Eldar, <a href="/A114636/b114636.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..2000 from Harvey P. Dale)

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

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

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

%F Numbers k such that A000567(k) is a term of A046310.

%F Numbers k such that A001222(A000567(k)) = 8.

%F Numbers k such that A001222(k) + A001222(3*k-2) = 8.

%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 A046310.

%e a(1) = 22 because OctagonalNumber(22) = Oct(22) = 22*(3*22-2) = 1408 = 2^7 * 11 has exactly 8 prime factors (seven are all equally 2; factors need not be distinct).

%e a(2) = 70 because Oct(70) = 70*(3*70-2) = 14560 = 2^5 * 5 * 7 * 13 is 8-almost prime.

%e a(3) = 80 because Oct(80) = 80*(3*80-2) = 19040 = 2^5 * 5 * 7 * 17.

%t Select[Range[400],PrimeOmega[PolygonalNumber[8,#]]==8&] (* _Harvey P. Dale_, Aug 31 2020 *)

%Y Cf. A000040, A000567, A001222, A001358, A014612, A014613, A014614, A046306, A046308, A046310.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Feb 18 2006