A114439 - OEIS

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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

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OFFSET

1,1

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).

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Pentagonal Number.

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}.

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

Cf. A000326, A001222, A001358, A115709.

Sequence in context: A327223 A143833 A186808 * A310570 A271146 A347806

Adjacent sequences: A114436 A114437 A114438 * A114440 A114441 A114442