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%I #10 Oct 02 2021 18:36:10
%S 4,5,6,10,13,14,29,34,38,41,46,53,58,73,86,94,101,106,109,118,134,149,
%T 181,206,214,218,226,233,254,274,281,293,314,326,349,394,398,401,409,
%U 421,449,454,458,461,478,538,541,566,569,613,626,634,661,673,694
%N Indices of semiprime pentagonal numbers.
%C 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 Harvey P. Dale, <a href="/A114439/b114439.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal Number.</a>
%F {a(n)} = {k such that A001222(A000326(k)) = 2}. {a(n)} = {k such that k*(3*k-1)/2 has exactly 2 prime factors}. {a(n)} = {k such that A000326(k) is an element of A001358}.
%e a(1) = 4 because P(4) = PentagonalNumber(4) = 4*(3*4 -1)/2 = 22 = 2 * 11 is semiprime.
%e a(2) = 5 because P(5) = 5*(3*5 -1)/2 = 35 = 5 * 7 is semiprime.
%e a(7) = 29 because P(29) = 29*(3*29 -1)/2 = 1247 = 29 * 43 is semiprime.
%e a(8) = 34 because P(34) = 34*(3*34 -1)/2 = 1717 = 17 * 101 is semiprime.
%e a(17) = 101 because P(101) = 101*(3*101 -1)/2 = 15251 = 101 * 151 is semiprime.
%t Position[PolygonalNumber[5,Range[700]],_?(PrimeOmega[#]==2&)]//Flatten (* _Harvey P. Dale_, Oct 02 2021 *)
%Y Cf. A000326, A001222, A001358, A115709.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Feb 14 2006
%E More terms from _Giovanni Resta_, Jun 14 2016
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