

A007304


Sphenic numbers: products of 3 distinct primes.
(Formerly M5207)


101



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

1,1


COMMENTS

Note the distinctions between this and "n has exactly three prime factors" (A014612) or "n has exactly three distinct prime factors." (A033992). 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. 3D analogue of 2D A037074 Product of twin primes, per Cino Hilliard's comment. Compare with 3D A107768 Golden 3almost 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.  JuriStepan 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

A008683(a(n)) = 1.
A000005(a(n)) = 8. [R. J. Mathar, Aug 14 2009]
A002033(a(n)1) = 13. [From JuriStepan 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 !! (n1)
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,changed


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



