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%I #16 Sep 08 2022 08:45:23
%S 64,96,128,144,162,182,198,216,224,234,246,270,278,288,304,310,320,
%T 324,352,390,414,416,432,438,480,504,528,544,550,558,584,594,600,646,
%U 648,654,662,684,694
%N Numbers n such that n-th octagonal number is 9-almost prime.
%C It is necessary but not sufficient that n must be prime (A000040), semiprime (A001358), 3-almost prime (A014612), 4-almost prime (A014613), 5-almost prime (A014614), 6-almost prime (A046306), 7-almost prime (A046308), or 8-almost prime (A046310).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OctagonalNumber.html">Octagonal Number.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlmostPrime.html">Almost Prime.</a>
%F n such that n*(3*n-2) has exactly nine prime factors (with multiplicity). n such that A000567(n) is an element of A046312. n such that A001222(A000567(n)) = 9. n such that A001222(n) + A001222(3*n-2) = 9. n such that [(3*n-2)*(3*n-1)*(3*n)]/[(3*n-2)+(3*n-1)+(3*n)] is an element of A046310.
%e 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 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 a(3) = 128 because Oct(128) = 128*(3*128-2) = 48896 = 2^8 * 191.
%o (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
%Y Cf. A000040, A000567, A001222, A001358, A014612, A014613, A014614, A046306, A046308, A046310, A046312, A088878, A114606, A114618, A114621, A114634, A114635, A114636.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Feb 19 2006
%E Missing terms inserted - R. J. Mathar, Dec 22 2010
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