|
%I #9 Aug 13 2021 15:41:15
%S 24,30,32,38,48,66,72,78,90,94,104,110,112,114,120,136,140,154,164,
%T 166,168,176,180,190,204,206,208,210,220,222,228,238,248,254,276,280,
%U 284,286,290,300,306,312,326,338,344
%N Numbers n such that n-th octagonal number is 7-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), or 6-almost prime (A046308).
%H Harvey P. Dale, <a href="/A114635/b114635.txt">Table of n, a(n) for n = 1..1000</a>
%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 seven prime factors (with multiplicity). n such that A000567(n) is an element of A046308. n such that A001222(A000567(n)) = 7. n such that A001222(n) + A001222(3*n-2) = 7. n such that [(3*n-2)*(3*n-1)*(3*n)]/[(3*n-2)+(3*n-1)+(3*n)] is an element of A046308.
%e a(1) = 24 because OctagonalNumber(24) = Oct(24) = 24*(3*24-2) = 96 = 1680 = 2^4 * 3 * 5 * 7 has exactly 7 prime factors (four are all equally 2; factors need not be distinct).
%e a(2) = 30 because Oct(30) = 30*(3*30-2) = 2640 = 2^4 * 3 * 5 * 11 is 7-almost prime.
%e a(3) = 32 because Oct(32) = 32*(3*32-2) = 3008 = 2^6 * 47 is 7-almost prime.
%t Select[Range[400],PrimeOmega[PolygonalNumber[8,#]]==7&] (* _Harvey P. Dale_, Aug 13 2021 *)
%Y Cf. A000040, A000567, A001222, A001358, A014612, A014613, A014614, A046306, A046308.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Feb 18 2006
|
