This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A114634 #22 Feb 16 2025 08:32:59
%S A114634 6,14,16,18,34,36,40,42,44,46,50,52,56,60,62,74,88,98,100,122,124,130,
%T A114634 132,135,138,142,148,152,156,158,170,178,186,189,194,196,209,226,232,
%U A114634 242,243,244,258,260,266,274,282,292,296,297,302,308,314,315,316,322
%N A114634 Numbers k such that the k-th octagonal number is 6-almost prime.
%C A114634 It is necessary but not sufficient that k must be prime (A000040), semiprime (A001358), 3-almost prime (A014612), 4-almost prime (A014613), or 5-almost prime (A014614).
%H A114634 Amiram Eldar, <a href="/A114634/b114634.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harvey P. Dale)
%H A114634 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>.
%H A114634 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OctagonalNumber.html">Octagonal Number</a>.
%F A114634 Numbers k such that k*(3*k-2) has exactly six prime factors (with multiplicity).
%F A114634 Numbers k such that A000567(k) is a term of A046306.
%F A114634 Numbers k such that A001222(A000567(k)) = 6.
%F A114634 Numbers k such that A001222(k) + A001222(3*k-2) = 6.
%F A114634 Numbers k such that [(3*k-2)*(3*k-1)*(3*k)]/[(3*k-2)+(3*k-1)+(3*k)] is a term of A046306.
%e A114634 a(1) = 6 because OctagonalNumber(6) = Oct(6) = 6*(3*6-2) = 96 = 2^5 * 3 has exactly 6 prime factors (five are all equally 2; factors need not be distinct).
%e A114634 a(2) = 14 because Oct(14) = 14*(3*14-2) = 560 = 2^4 * 5 * 7 is 6-almost prime.
%e A114634 a(3) = 16 because Oct(16) = 16*(3*16-2) = 736 = 2^5 * 23.
%e A114634 a(7) = 40 because Oct(40) = 40*(3*40-2) = 4720 = 2^4 * 5 * 59 [also, 4720 = Oct(40) = Oct(Oct(4)), an iterated octagonal number].
%e A114634 a(19) = 100 because Oct(100) = 100*(3*100-2) = 29800 = 2^3 * 5^2 * 149.
%t A114634 Flatten[Position[Table[n(3n-2),{n,400}],_?(PrimeOmega[#]==6&)]] (* _Harvey P. Dale_, Jun 17 2013 *)
%t A114634 Select[Range[400],PrimeOmega[PolygonalNumber[8,#]]==6&] (* _Harvey P. Dale_, Feb 23 2022 *)
%o A114634 (PARI) is(n)=my(t=bigomega(3*n-2)); t<6 && (t<5 || !isprime(n)) && t+bigomega(n)==6 \\ _Charles R Greathouse IV_, Feb 01 2017
%Y A114634 Cf. A000040, A000567, A001222, A001358, A014612, A014613, A014614, A046306.
%K A114634 easy,nonn
%O A114634 1,1
%A A114634 _Jonathan Vos Post_, Feb 17 2006