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A007304
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Products of 3 distinct primes.
(Formerly M5207)
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91
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30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 345, 354, 357, 366, 370, 374, 385, 399, 402, 406, 410, 418, 426, 429, 430, 434, 435, 438
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OFFSET
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1,1
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COMMENTS
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Also called sphenic numbers. Moebius function of n is -1. Note the distinctions between this and "n has exactly three prime factors" or "n has exactly three distinct prime factors." The word "sphenic" also means "shaped like a wedge" [American Heritage Dictionary] as in dentation with "sphenic molars." - Jonathan Vos Post, Sep 11 2005
Also the volume of a sphenic brick. A sphenic brick is a rectangular parallelopiped whose sides are components of a sphenic number, namely whose sides are three distinct primes. Example: The distinct prime triple (3,5,7) produces a 3x5x7 unit brick which has volume 105 cubic units. 3-D analogue of 2-D A037074 Product of twin primes, per Cino Hilliard's comment. Compare with 3-D A107768 Golden 3-almost primes = Volumes of bricks (rectangular parallelopipeds) each of whose faces has golden semiprime area. - Jonathan Vos Post, Jan 08 2007
Or the numbers n such that 13 = number of perfect partitions of n. - Juri-Stepan Gerasimov, Oct 07 2009
A178254(a(n)) = 36. [From Reinhard Zumkeller, May 24 2010]
Sum(n>=1, 1/a(n)^s) = (1/6)*(P(s)^3 - P(3*s) - 3*(P(s)*P(2*s)-P(3*s))), where P is prime Zeta function. - Enrique Pérez Herrero, Jun 28 2012
Also numbers n with A001222(n)=3 and A001221(n)=3. - Enrique Pérez Herrero, Jun 28 2012
A050326(a(n)) = 5, subsequence of A225228; A162143(n) = a(n)^2. - Reinhard Zumkeller, May 03 2013
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
"Sphenic", The American Heritage Dictionary of the English Language, Fourth Edition, Houghton Mifflin Company, 2000.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..10000
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FORMULA
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A000005(a(n)) = 8. [R. J. Mathar, Aug 14 2009]
A002033(a(n)-1) = 13. [From Juri-Stepan Gerasimov, Oct 07 2009, R. J. Mathar, Oct 14 2009]
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MAPLE
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a:=proc(n) if bigomega(n)=3 and nops(factorset(n))=3 then n else fi end: seq(a(n), n=1..450); (Emeric Deutsch)
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MATHEMATICA
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Union[Flatten[Table[Prime[n]*Prime[m]*Prime[k], {k, 20}, {n, k+1, 20}, {m, n+1, 20}]]]
Take[ Sort@ Flatten@ Table[ Prime@i Prime@j Prime@k, {i, 3, 21}, {j, 2, i - 1}, {k, j - 1}], 53] (* Robert G. Wilson v *)
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PROG
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(PARI) for(n=1, 1e4, if(bigomega(n)==3, print1(n", "))) \\ Charles R Greathouse IV, Jun 10 2011
(PARI) list(lim)=my(v=List(), t); forprime(p=2, (lim)^(1/3), forprime(q=p+1, sqrt(lim\p), t=p*q; forprime(r=q+1, lim\t, listput(v, t*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011
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CROSSREFS
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Cf. A006881, A046386, A046387, A067885 (product of 2, 4, 5 and 6 distinct primes, resp.)
Cf. A046389, A046393, A061299, A067467, A071140, A096917, A096918, A096919, A100765, A103653, A107464.
Cf. A037074, A107768.
Cf. A002033.
Cf. A179643, A179695.
Sequence in context: A090815 A225228 A093599 * A160350 A053858 A075819
Adjacent sequences: A007301 A007302 A007303 * A007305 A007306 A007307
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KEYWORD
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nonn,changed
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AUTHOR
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Simon Plouffe
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EXTENSIONS
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More terms from Robert G. Wilson v, Jan 04 2006
Comment concerning number of divisors corrected by R. J. Mathar, Aug 14 2009
Formula index corrected - R. J. Mathar, Oct 14 2009
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STATUS
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approved
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