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A114517 Numbers k such that the k-th heptagonal number is semiprime.

%I A114517 #22 Feb 16 2025 08:32:59
%S A114517 4,5,10,13,14,17,22,26,29,34,41,46,53,61,62,73,74,94,97,101,109,113,
%T A114517 118,122,146,158,166,173,178,194,197,218,229,241,257,262,274,277,281,
%U A114517 298,314,326,334,353,358,382,389,397,398,409,421,454,458,461,521,538
%N A114517 Numbers k such that the k-th heptagonal number is semiprime.
%C A114517 Hep(2) = 7 is the only prime heptagonal number.
%H A114517 Amiram Eldar, <a href="/A114517/b114517.txt">Table of n, a(n) for n = 1..10000</a>
%H A114517 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeptagonalNumber.html">Heptagonal Number</a>.
%H A114517 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Semiprime.html">Semiprime</a>.
%F A114517 Numbers k such that Hep(k) = k*(5*k-3)/2 is semiprime.
%F A114517 Numbers k such that A000566(k) is a term of A001358.
%F A114517 Numbers k such that A001222(A000566(k)) = 2.
%F A114517 Numbers k such that A001222(k*(5*k-3)/2) = 2.
%F A114517 Numbers k such that [k/2 is prime and 5*k-3 is prime] or [k is prime and (5*k-3)/2 is prime].
%e A114517 a(1) = 4 because Hep(4) = 4*(5*4-3)/2 = 34 = 2 * 17 is semiprime.
%e A114517 a(2) = 5 because Hep(5) = 5*(5*5-3)/2 = 55 = 5 * 11 is semiprime.
%e A114517 a(10) = 34 because Hep(34) = 2839 = 17 * 167 is semiprime and this is also the first iterated heptagonal semiprime Hep(34) = Hep(Hep(4)).
%e A114517 a(20) = 101 because Hep(101) = 25351 = 101 * 251 is semiprime [and brilliant].
%t A114517 Select[Range[700],PrimeOmega[(#(5#-3))/2]==2&] (* _Harvey P. Dale_, Jul 24 2011 *)
%Y A114517 Cf. A000040, A000566, A001222, A001358, A099153.
%K A114517 easy,nonn
%O A114517 1,1
%A A114517 _Jonathan Vos Post_, Feb 15 2006
%E A114517 More terms from _Harvey P. Dale_, Jul 24 2011