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Numbers k such that the k-th hexagonal number is a 3-almost prime.
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%I #17 Oct 06 2024 09:14:22

%S 4,5,6,9,10,11,13,15,17,21,22,29,34,43,47,49,51,55,57,61,67,69,71,73,

%T 82,87,89,91,101,103,106,107,109,115,121,127,129,141,142,151,159,166,

%U 169,177,181,187,191,197,201,205,217,223,227,241,251,262,269,274,277,283

%N Numbers k such that the k-th hexagonal number is a 3-almost prime.

%C There are no prime hexagonal numbers. The k-th hexagonal number A000384(k) = k*(2*k-1) is semiprime iff both k and 2*k-1 are primes iff A000384(k) is an element of A001358 iff k is an element of A005382.

%H Amiram Eldar, <a href="/A114454/b114454.txt">Table of n, a(n) for n = 1..10000</a>

%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/HexagonalNumber.html">Hexagonal Number</a>.

%F Numbers k such that hexagonal number A000384(k) is an element of A014612.

%F Numbers k such that A001222(A000384(k)) = 3.

%F Numbers k such that A001222(k*(2*k-1)) = 3.

%e a(1) = 4 because HexagonalNumber(4) = H(4) = 4*(2*4-1) = 28 = 2^2 * 7 is a 3-almost prime.

%e a(2) = 5 because H(5) = 5*(2*5-1) = 45 = 3^2 * 5 is a 3-almost prime.

%e a(3) = 6 because H(6) = 6*(2*6-1) = 66 = 2 * 3 * 11 is a 3-almost prime.

%p A000384 := proc(n) n*(2*n-1) ; end: isA014612 := proc(n) option remember ; RETURN( numtheory[bigomega](n) = 3) ; end: for n from 1 to 400 do if isA014612(A000384(n)) then printf("%d,",n) ; fi; od: # _R. J. Mathar_, Jan 27 2009

%t Select[Range[300], PrimeOmega[PolygonalNumber[6, #]] == 3 &] (* _Amiram Eldar_, Oct 06 2024 *)

%Y Cf. A000384, A001222, A014612.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Feb 14 2006

%E 151 to 177 inserted and extended by _R. J. Mathar_, Jan 27 2009