1,1

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).

Eric Weisstein's World of Mathematics, Pentagonal Number.

{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}.

a(1) = 4 because P(4) = PentagonalNumber(4) = 4*(3*4 -1)/2 = 22 = 2 * 11 is semiprime.

a(2) = 5 because P(5) = 5*(3*5 -1)/2 = 35 = 5 * 7 is semiprime.

a(7) = 29 because P(29) = 29*(3*29 -1)/2 = 1247 = 29 * 43 is semiprime.

a(8) = 34 because P(34) = 34*(3*34 -1)/2 = 1717 = 17 * 101 is semiprime.

a(17) = 101 because P(101) = 101*(3*101 -1)/2 = 15251 = 101 * 151 is semiprime.

Cf. A000326, A001222, A001358, A115709.

Sequence in context: A024565 A143833 A066501 this_sequence A079257 A001609 A101590

Adjacent sequences: A114436 A114437 A114438 this_sequence A114440 A114441 A114442

easy,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 14 2006