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

4, 5, 10, 13, 14, 17, 22, 26, 29, 34, 41, 46, 53, 61, 62, 73, 74, 94, 97, 101, 109, 113, 118, 122, 146, 158, 166, 173, 178, 194, 197, 218, 229, 241, 257, 262, 274, 277, 281, 298, 314, 326, 334, 353, 358, 382, 389, 397, 398, 409, 421, 454, 458, 461, 521, 538

Offset: 1

Author

Jonathan Vos Post, Feb 15 2006

Keywords

• easy
• nonn

Comments

Hep(2) = 7 is the only prime heptagonal number.

Examples

			a(1) = 4 because Hep(4) = 4*(5*4-3)/2 = 34 = 2 * 17 is semiprime.
a(2) = 5 because Hep(5) = 5*(5*5-3)/2 = 55 = 5 * 11 is semiprime.
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)).
a(20) = 101 because Hep(101) = 25351 = 101 * 251 is semiprime [and brilliant].
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Heptagonal Number.
• Eric Weisstein's World of Mathematics, Semiprime.

Crossrefs

Cf. A000040, A000566, A001222, A001358, A099153.

Programs

Mathematica

Select[Range[700],PrimeOmega[(#(5#-3))/2]==2&] (* Harvey P. Dale, Jul 24 2011 *)

Formula

Numbers k such that Hep(k) = k*(5*k-3)/2 is semiprime.

Numbers k such that A000566(k) is a term of A001358.

Numbers k such that A001222(A000566(k)) = 2.

Numbers k such that A001222(k*(5*k-3)/2) = 2.

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].

Extensions

More terms from Harvey P. Dale, Jul 24 2011