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A113432 Pierpont semiprimes: semiprimes of the form (2^K)*(3^L)+1.

%I A113432 #19 Feb 16 2025 08:32:59
%S A113432 4,9,10,25,33,49,55,65,82,129,145,217,289,649,865,973,1537,1945,2049,
%T A113432 2305,3073,4097,4609,5833,6145,6913,8193,8749,9217,11665,13123,15553,
%U A113432 20737,23329,24577,27649,31105,34993,41473,62209,69985,73729,78733
%N A113432 Pierpont semiprimes: semiprimes of the form (2^K)*(3^L)+1.
%H A113432 Vincenzo Librandi, <a href="/A113432/b113432.txt">Table of n, a(n) for n = 1..89</a>
%H A113432 Chris Caldwell, <a href="http://groups.yahoo.com/group/primeform/message/6588/">"Pierpont primes." primeform posting, Oct 25, 2005.</a>
%H A113432 Chris Caldwell, <a href="/A113432/a113432.pdf">"Pierpont primes." primeform posting, Oct 25, 2005.</a> [Cached copy]
%H A113432 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PierpontPrime.html">Pierpont Prime</a>
%H A113432 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Semiprime.html">Semiprime</a>
%F A113432 {a(n)} = Intersection of {(2^K)*(3^L)+1} A055600 and semiprimes A001358. a(n) is in this sequence iff there exist nonnegative integers K and L such that Omega((2^K)*(3^L)+1) = 2.
%e A113432 a(1) = 4 = (2^0)*(3^1)+1 = 2^2 hence the semiprime A001358(1).
%e A113432 a(2) = 9 = (2^3)*(3^0)+1 = 3^2 hence the semiprime A001358(3).
%e A113432 a(3) = 10 = (2^0)*(3^2)+1 = 2 * 5 hence the semiprime A001358(4).
%e A113432 a(4) = 25 = (2^3)*(3^1)+1 = 5^2 hence the semiprime A001358(9).
%e A113432 a(5) = 33 = (2^5)*(3^0)+1 = 3 * 11 hence the semiprime A001358(11).
%e A113432 a(6) = 49 = (2^4)*(3^1)+1 = 7^2 hence the semiprime A001358(17).
%e A113432 a(7) = 55 = (2^1)*(3^3)+1 = 5 * 11 hence the semiprime A001358(19).
%t A113432 Select[Range[10^5], Plus @@ Last /@ FactorInteger[ # ] == 2 && Max @@ First /@ FactorInteger[ # - 1] < 5 &] (* _Ray Chandler_, Jan 24 2006 *)
%Y A113432 Cf. A001358, A003586, A005109, A055600, A111153, A111206, A113433, A113434.
%K A113432 easy,nonn
%O A113432 1,1
%A A113432 _Jonathan Vos Post_, Nov 01 2005