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A114828 Numbers k such that the k-th octagonal number has 9 prime factors counted with multiplicity.

%I A114828 #37 Feb 16 2025 08:33:00
%S A114828 64,96,128,144,162,182,198,216,224,234,246,270,278,288,304,310,320,
%T A114828 324,352,390,414,416,432,438,480,504,528,544,550,558,584,594,600,646,
%U A114828 648,654,662,684,694,702,710,729,750,752,756,798,810,834,850,870,888,900
%N A114828 Numbers k such that the k-th octagonal number has 9 prime factors counted with multiplicity.
%C A114828 k has at most 8 prime factors counted with multiplicity.
%H A114828 Amiram Eldar, <a href="/A114828/b114828.txt">Table of n, a(n) for n = 1..10000</a>
%H A114828 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>.
%H A114828 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OctagonalNumber.html">Octagonal Number</a>.
%F A114828 Integers k such that k*(3*k-2) has exactly nine prime factors (with multiplicity).
%F A114828 Integers k such that A000567(k) is a term of A046312.
%F A114828 Integers k such that A001222(A000567(k)) = 9.
%F A114828 Integers k such that A001222(k) + A001222(3*k-2) = 9.
%F A114828 Integers k such that (3*k-2)*(3*k-1)*(3*k)/((3*k-2)+(3*k-1)+(3*k)) is in A046310.
%e A114828 a(1) = 64 because OctagonalNumber(64) = Oct(64) = 64*(3*64-2) = 12160 = 2^7 * 5 * 19 has exactly 9 prime factors (seven are all equally 2; factors need not be distinct).
%e A114828 a(2) = 96 because Oct(96) = 96*(3*96-2) = 27456 = 2^6 * 3 * 11 * 13 is 9-almost prime [also 27456 = Oct(96) = Oct(Oct(6)) is an iterated octagonal number].
%e A114828 a(3) = 128 because Oct(128) = 128*(3*128-2) = 48896 = 2^8 * 191.
%t A114828 Select[Range[900],PrimeOmega[PolygonalNumber[8,#]]==9&] (* _James C. McMahon_, Jul 30 2024 *)
%o A114828 (Magma) A000567:=func< n | n*(3*n-2) >; Is9almostprime:=func< n | &+[k[2]: k in Factorization(n)] eq 9 >; [ n: n in [2..1000] | Is9almostprime(A000567(n)) ]; // _Klaus Brockhaus_, Dec 22 2010
%o A114828 (PARI) isok(k) = bigomega(k*(3*k-2)) == 9; \\ _Michel Marcus_, Aug 02 2024
%Y A114828 Cf. A000040, A000567, A001222, A001358, A014612, A014613, A014614, A046306, A046308, A046310, A046312, A088878, A114606, A114618, A114621, A114634, A114635, A114636.
%K A114828 easy,nonn
%O A114828 1,1
%A A114828 _Jonathan Vos Post_, Feb 19 2006
%E A114828 Missing terms inserted by _R. J. Mathar_, Dec 22 2010
%E A114828 a(40)-a(52) from _James C. McMahon_, Jul 30 2024
%E A114828 Name edited by _David A. Corneth_, Jul 31 2024