A114439 Indices of semiprime pentagonal numbers.

4, 5, 6, 10, 13, 14, 29, 34, 38, 41, 46, 53, 58, 73, 86, 94, 101, 106, 109, 118, 134, 149, 181, 206, 214, 218, 226, 233, 254, 274, 281, 293, 314, 326, 349, 394, 398, 401, 409, 421, 449, 454, 458, 461, 478, 538, 541, 566, 569, 613, 626, 634, 661, 673, 694

Offset: 1

Author

Jonathan Vos Post, Feb 14 2006

Keywords

• easy
• nonn

Comments

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).

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
• Eric Weisstein's World of Mathematics, Pentagonal Number.

Crossrefs

Cf. A000326, A001222, A001358, A115709.

Programs

Mathematica

Position[PolygonalNumber[5,Range[700]],?(PrimeOmega[#]==2&)]//Flatten (* _Harvey P. Dale, Oct 02 2021 *)

Formula

{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}.

Extensions

More terms from Giovanni Resta, Jun 14 2016