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A114439
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Indices of semiprime pentagonal numbers.
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1
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4, 5, 6, 10, 13, 14, 29, 34, 38, 41, 46, 53, 58, 73, 86, 94, 101, 106, 109, 118, 134, 149, 181
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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).
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LINKS
| Eric Weisstein's World of Mathematics, Pentagonal Number.
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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}.
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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.
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CROSSREFS
| Cf. A000326, A001222, A001358, A115709.
Sequence in context: A143833 A066501 A186808 * A079257 A001609 A101590
Adjacent sequences: A114436 A114437 A114438 * A114440 A114441 A114442
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KEYWORD
| easy,nonn
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 14 2006
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