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A007304 Products of 3 distinct primes.
(Formerly M5207)
99
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

REFERENCES

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.

LINKS

T. D. Noe, Table of n, a(n) for n=1..10000

FORMULA

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]

MAPLE

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)

MATHEMATICA

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 *)

PROG

(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

(Haskell)

a007304 n = a007304_list !! (n-1)

a007304_list = filter f [1..] where

   f u = p < q && q < w && a010051 w == 1 where

         p = a020639 u; v = div u p; q = a020639 v; w = div v q

-- Reinhard Zumkeller, Mar 23 2014

CROSSREFS

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.

Cf. A020639, A010051, A239656 (first differences).

Sequence in context: A238367 A225228 A093599 * A160350 A053858 A075819

Adjacent sequences:  A007301 A007302 A007303 * A007305 A007306 A007307

KEYWORD

nonn

AUTHOR

Simon Plouffe

EXTENSIONS

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

STATUS

approved

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Last modified April 21 02:07 EDT 2014. Contains 240824 sequences.