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A114446
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Indices of 7-almost prime pentagonal numbers.
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1
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27, 43, 96, 107, 128, 147, 180, 187, 203, 224, 288, 312, 336, 352, 360, 387, 392, 395, 400, 411, 416, 475, 480, 486, 491, 495, 523, 539, 544, 560, 572, 587, 592, 600, 603, 619, 621, 627, 635, 704, 729, 735, 752, 763, 779, 795, 800, 810, 819, 840, 843, 882
(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].
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LINKS
| Eric Weisstein's World of Mathematics, Pentagonal Number.
Eric Weisstein's World of Mathematics, Almost Prime.
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FORMULA
| {a(n)} = {k such that A001222(A000326(k)) = 7}. {a(n)} = {k such that k*(3*k-1)/2 has exactly 7 prime factors}. {a(n)} = {k such that A000326(k) is an element of A046308}.
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EXAMPLE
| a(1) = 27 because P(27) = PentagonalNumber(27) = 27*(3*27-1)/2 = 1080 = 2^3 * 3^3 * 5 is a 7-almost prime.
a(2) = 43 because P(43) = 43*(3*43-1)/2 = 2752 = 2^6 * 43 is a 7-almost
prime.
a(7) = 180 because P(180) = 180*(3*180-1)/2 = 48510 = 2 * 3^2 * 5 x 7^2 * 11 is a 7-almost prime.
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MATHEMATICA
| Select[Range[2000], PrimeOmega[# (3#-1)/2]==7&] (* From Harvey P. Dale, Jul 16 2011 *)
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CROSSREFS
| Cf. A000326, A001222, A046308.
Sequence in context: A181489 A117103 A124940 * A141229 A121614 A046340
Adjacent sequences: A114443 A114444 A114445 * A114447 A114448 A114449
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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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EXTENSIONS
| More terms from Harvey P. Dale, Jul 16 2011
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