cp's OEIS Frontend

A114439 Indices of semiprime pentagonal numbers.

%I A114439 #19 Feb 16 2025 08:32:59
%S A114439 4,5,6,10,13,14,29,34,38,41,46,53,58,73,86,94,101,106,109,118,134,149,
%T A114439 181,206,214,218,226,233,254,274,281,293,314,326,349,394,398,401,409,
%U A114439 421,449,454,458,461,478,538,541,566,569,613,626,634,661,673,694
%N A114439 Indices of semiprime pentagonal numbers.
%C A114439 P(2) = 5 is the only prime pentagonal number, all other factor as P(k) = (k/2)*(3*k-1) or k*((3*k-1)/2) and thus have at least 2 prime factors. P(k) is semiprime iff [k prime and (3*k-1)/2 prime] or [k/2 prime and 3*k-1 prime]. A115709 is pentagonal numbers (A000326) whose digit reversal is a semiprime (A001358).
%H A114439 Amiram Eldar, <a href="/A114439/b114439.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harvey P. Dale)
%H A114439 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal Number</a>.
%F A114439 {a(n)} = {k such that A001222(A000326(k)) = 2}.
%F A114439 {a(n)} = {k such that k*(3*k-1)/2 has exactly 2 prime factors}.
%F A114439 {a(n)} = {k such that A000326(k) is an element of A001358}.
%e A114439 a(1) = 4 because P(4) = PentagonalNumber(4) = 4*(3*4 -1)/2 = 22 = 2 * 11 is semiprime.
%e A114439 a(2) = 5 because P(5) = 5*(3*5 -1)/2 = 35 = 5 * 7 is semiprime.
%e A114439 a(7) = 29 because P(29) = 29*(3*29 -1)/2 = 1247 = 29 * 43 is semiprime.
%e A114439 a(8) = 34 because P(34) = 34*(3*34 -1)/2 = 1717 = 17 * 101 is semiprime.
%e A114439 a(17) = 101 because P(101) = 101*(3*101 -1)/2 = 15251 = 101 * 151 is semiprime.
%t A114439 Position[PolygonalNumber[5,Range[700]],_?(PrimeOmega[#]==2&)]//Flatten (* _Harvey P. Dale_, Oct 02 2021 *)
%Y A114439 Cf. A000326, A001222, A001358, A115709.
%K A114439 easy,nonn
%O A114439 1,1
%A A114439 _Jonathan Vos Post_, Feb 14 2006
%E A114439 More terms from _Giovanni Resta_, Jun 14 2016