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Numbers k such that the k-th octagonal number is 3-almost prime.
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%I #17 Oct 07 2024 01:15:49

%S 2,15,17,19,21,25,29,31,33,35,41,51,55,65,73,77,79,83,89,91,93,95,97,

%T 101,107,111,123,131,133,139,141,145,149,151,155,157,173,179,183,197,

%U 201,203,205,215,221,223,227,229,233,237,241,247,253

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

%C It is necessary but not sufficient that k must be either prime or semiprime.

%H Amiram Eldar, <a href="/A114606/b114606.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/OctagonalNumber.html">Octagonal Number</a>.

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

%F Numbers k such that A000567(k) is a term of A014612.

%F Numbers k such that A001222(A000567(k)) = 3.

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

%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 A014612.

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

%e a(2) = 15 because Oct(15) = 15*(3*15-2) = 645 = 3 * 5 * 43, a 3-almost prime.

%e a(5) = 21 because Oct(21) = 21*(3*21-2) = 1281 = 3 * 7 * 61 [also, 1281 = Oct(21) = Oct(Oct(3)) is an iterated octagonal number].

%e a(14) = 65 because Oct(65) = 65*(3*65-2) = 12545 = 5 * 13 * 193 [also, 12545 = Oct(65) = Oct(Oct(5)) is an iterated octagonal number].

%e a(29) = 133 because Oct(133) = 133*(3*133-2) = 52801 = 7 * 19 * 397 [also, 52801 = Oct(133) = Oct(Oct(7)) is an iterated octagonal number].

%p A000567 := proc(n) n*(3*n-2) ; end: isA014612 := proc(n) RETURN( numtheory[bigomega](n) = 3) ; end: for n from 1 to 1000 do q := A000567(n) ; if isA014612(q) then printf("%d,",n) ; fi; od: # _R. J. Mathar_, Jan 27 2009

%t Select[Range[500], PrimeOmega[PolygonalNumber[8, #]] == 3 &] (* _Amiram Eldar_, Oct 07 2024 *)

%Y Cf. A000040, A000567, A001222, A014612.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Feb 17 2006